[{"data":1,"prerenderedAt":16307},["ShallowReactive",2],{"note:math\u002Fabstract_algebra\u002Fgroupproblem":3,"site-search-catalogue":16306},{"id":4,"title":5,"body":6,"description":16290,"extension":16298,"meta":16299,"navigation":16301,"path":16302,"seo":16303,"stem":16304,"__hash__":16305},"notes\u002Fsource\u002Fmath\u002Fabstract_algebra\u002Fgroupproblem.md","特殊类型的群 问题详解",{"type":7,"value":8,"toc":16289},"minimark",[9,13,16,21,26,455,460,3901,3903,3907,3968,3972,6540,6542,6691,6894,6968,7192,7342,8079,8152,8285,8515,8517,8521,8525,8862,8867,11010,11012,11141,11284,11289,14563,14565,14569,14690,14694],[10,11,5],"h1",{"id":12},"特殊类型的群-问题详解",[14,15],"hr",{},[17,18,20],"h3",{"id":19},"第一部分循环群与生成元基础篇","第一部分：循环群与生成元（基础篇）",[22,23,25],"h4",{"id":24},"题目-2方程解的唯一性","题目 2：方程解的唯一性",[27,28,29,33,34,80,81,112,113,184,185,216,217,316,317,412,413,454],"p",{},[30,31,32],"strong",{},"题目："," 如果群 ",[35,36,39,61],"span",{"className":37},[38],"katex",[35,40,43],{"className":41},[42],"katex-mathml",[44,45,47],"math",{"xmlns":46},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[48,49,50,57],"semantics",{},[51,52,53],"mrow",{},[54,55,56],"mi",{},"G",[58,59,56],"annotation",{"encoding":60},"application\u002Fx-tex",[35,62,66],{"className":63,"ariaHidden":65},[64],"katex-html","true",[35,67,70,75],{"className":68},[69],"base",[35,71],{"className":72,"style":74},[73],"strut","height:0.6833em;",[35,76,56],{"className":77},[78,79],"mord","mathnormal"," 中的元素 ",[35,82,84,98],{"className":83},[38],[35,85,87],{"className":86},[42],[44,88,89],{"xmlns":46},[48,90,91,96],{},[51,92,93],{},[54,94,95],{},"g",[58,97,95],{"encoding":60},[35,99,101],{"className":100,"ariaHidden":65},[64],[35,102,104,108],{"className":103},[69],[35,105],{"className":106,"style":107},[73],"height:0.625em;vertical-align:-0.1944em;",[35,109,95],{"className":110,"style":111},[78,79],"margin-right:0.0359em;"," 的阶 ",[35,114,116,143],{"className":115},[38],[35,117,119],{"className":118},[42],[44,120,121],{"xmlns":46},[48,122,123,140],{},[51,124,125,129,131,133,137],{},[54,126,128],{"mathvariant":127},"normal","∣",[54,130,95],{},[54,132,128],{"mathvariant":127},[134,135,136],"mo",{},"=",[54,138,139],{},"n",[58,141,142],{"encoding":60},"|g| = n",[35,144,146,174],{"className":145,"ariaHidden":65},[64],[35,147,149,153,156,159,162,167,171],{"className":148},[69],[35,150],{"className":151,"style":152},[73],"height:1em;vertical-align:-0.25em;",[35,154,128],{"className":155},[78],[35,157,95],{"className":158,"style":111},[78,79],[35,160,128],{"className":161},[78],[35,163],{"className":164,"style":166},[165],"mspace","margin-right:0.2778em;",[35,168,136],{"className":169},[170],"mrel",[35,172],{"className":173,"style":166},[165],[35,175,177,181],{"className":176},[69],[35,178],{"className":179,"style":180},[73],"height:0.4306em;",[35,182,139],{"className":183},[78,79]," 与正整数 ",[35,186,188,202],{"className":187},[38],[35,189,191],{"className":190},[42],[44,192,193],{"xmlns":46},[48,194,195,200],{},[51,196,197],{},[54,198,199],{},"k",[58,201,199],{"encoding":60},[35,203,205],{"className":204,"ariaHidden":65},[64],[35,206,208,212],{"className":207},[69],[35,209],{"className":210,"style":211},[73],"height:0.6944em;",[35,213,199],{"className":214,"style":215},[78,79],"margin-right:0.0315em;"," 互素（即 ",[35,218,220,258],{"className":219},[38],[35,221,223],{"className":222},[42],[44,224,225],{"xmlns":46},[48,226,227,255],{},[51,228,229,232,235,239,241,244,246,249,251],{},[54,230,231],{},"gcd",[134,233,234],{},"⁡",[134,236,238],{"stretchy":237},"false","(",[54,240,139],{},[134,242,243],{"separator":65},",",[54,245,199],{},[134,247,248],{"stretchy":237},")",[134,250,136],{},[252,253,254],"mn",{},"1",[58,256,257],{"encoding":60},"\\gcd(n, k) = 1",[35,259,261,306],{"className":260,"ariaHidden":65},[64],[35,262,264,267,275,279,282,286,290,293,297,300,303],{"className":263},[69],[35,265],{"className":266,"style":152},[73],[35,268,271,274],{"className":269},[270],"mop",[35,272,95],{"style":273},"margin-right:0.0139em;","cd",[35,276,238],{"className":277},[278],"mopen",[35,280,139],{"className":281},[78,79],[35,283,243],{"className":284},[285],"mpunct",[35,287],{"className":288,"style":289},[165],"margin-right:0.1667em;",[35,291,199],{"className":292,"style":215},[78,79],[35,294,248],{"className":295},[296],"mclose",[35,298],{"className":299,"style":166},[165],[35,301,136],{"className":302},[170],[35,304],{"className":305,"style":166},[165],[35,307,309,313],{"className":308},[69],[35,310],{"className":311,"style":312},[73],"height:0.6444em;",[35,314,254],{"className":315},[78],"），证明方程 ",[35,318,320,344],{"className":319},[38],[35,321,323],{"className":322},[42],[44,324,325],{"xmlns":46},[48,326,327,341],{},[51,328,329,337,339],{},[330,331,332,335],"msup",{},[54,333,334],{},"x",[54,336,199],{},[134,338,136],{},[54,340,95],{},[58,342,343],{"encoding":60},"x^k = g",[35,345,347,403],{"className":346,"ariaHidden":65},[64],[35,348,350,354,394,397,400],{"className":349},[69],[35,351],{"className":352,"style":353},[73],"height:0.8491em;",[35,355,357,360],{"className":356},[78],[35,358,334],{"className":359},[78,79],[35,361,364],{"className":362},[363],"msupsub",[35,365,368],{"className":366},[367],"vlist-t",[35,369,372],{"className":370},[371],"vlist-r",[35,373,376],{"className":374,"style":353},[375],"vlist",[35,377,379,384],{"style":378},"top:-3.063em;margin-right:0.05em;",[35,380],{"className":381,"style":383},[382],"pstrut","height:2.7em;",[35,385,391],{"className":386},[387,388,389,390],"sizing","reset-size6","size3","mtight",[35,392,199],{"className":393,"style":215},[78,79,390],[35,395],{"className":396,"style":166},[165],[35,398,136],{"className":399},[170],[35,401],{"className":402,"style":166},[165],[35,404,406,409],{"className":405},[69],[35,407],{"className":408,"style":107},[73],[35,410,95],{"className":411,"style":111},[78,79]," 在 ",[35,414,416,436],{"className":415},[38],[35,417,419],{"className":418},[42],[44,420,421],{"xmlns":46},[48,422,423,433],{},[51,424,425,428,430],{},[134,426,427],{"stretchy":237},"⟨",[54,429,95],{},[134,431,432],{"stretchy":237},"⟩",[58,434,435],{"encoding":60},"\\langle g \\rangle",[35,437,439],{"className":438,"ariaHidden":65},[64],[35,440,442,445,448,451],{"className":441},[69],[35,443],{"className":444,"style":152},[73],[35,446,427],{"className":447},[278],[35,449,95],{"className":450,"style":111},[78,79],[35,452,432],{"className":453},[296]," 内恰有一个解。",[27,456,457],{},[30,458,459],{},"【从零解析】",[461,462,463,919],"ol",{},[464,465,466,469,470],"li",{},[30,467,468],{},"背景知识","：",[471,472,473,723],"ul",{},[464,474,475,513,514,542,543,722],{},[35,476,478,495],{"className":477},[38],[35,479,481],{"className":480},[42],[44,482,483],{"xmlns":46},[48,484,485,493],{},[51,486,487,489,491],{},[134,488,427],{"stretchy":237},[54,490,95],{},[134,492,432],{"stretchy":237},[58,494,435],{"encoding":60},[35,496,498],{"className":497,"ariaHidden":65},[64],[35,499,501,504,507,510],{"className":500},[69],[35,502],{"className":503,"style":152},[73],[35,505,427],{"className":506},[278],[35,508,95],{"className":509,"style":111},[78,79],[35,511,432],{"className":512},[296]," 是由 ",[35,515,517,530],{"className":516},[38],[35,518,520],{"className":519},[42],[44,521,522],{"xmlns":46},[48,523,524,528],{},[51,525,526],{},[54,527,95],{},[58,529,95],{"encoding":60},[35,531,533],{"className":532,"ariaHidden":65},[64],[35,534,536,539],{"className":535},[69],[35,537],{"className":538,"style":107},[73],[35,540,95],{"className":541,"style":111},[78,79]," 生成的循环群，里面的元素长这样：",[35,544,546,600],{"className":545},[38],[35,547,549],{"className":548},[42],[44,550,551],{"xmlns":46},[48,552,553,597],{},[51,554,555,558,561,563,565,567,574,576,579,581,594],{},[134,556,557],{"stretchy":237},"{",[54,559,560],{},"e",[134,562,243],{"separator":65},[54,564,95],{},[134,566,243],{"separator":65},[330,568,569,571],{},[54,570,95],{},[252,572,573],{},"2",[134,575,243],{"separator":65},[134,577,578],{},"…",[134,580,243],{"separator":65},[330,582,583,585],{},[54,584,95],{},[51,586,587,589,592],{},[54,588,139],{},[134,590,591],{},"−",[252,593,254],{},[134,595,596],{"stretchy":237},"}",[58,598,599],{"encoding":60},"\\{e, g, g^2, \\dots, g^{n-1}\\}",[35,601,603],{"className":602,"ariaHidden":65},[64],[35,604,606,610,613,616,619,622,625,628,631,661,664,667,671,674,677,680,719],{"className":605},[69],[35,607],{"className":608,"style":609},[73],"height:1.0641em;vertical-align:-0.25em;",[35,611,557],{"className":612},[278],[35,614,560],{"className":615},[78,79],[35,617,243],{"className":618},[285],[35,620],{"className":621,"style":289},[165],[35,623,95],{"className":624,"style":111},[78,79],[35,626,243],{"className":627},[285],[35,629],{"className":630,"style":289},[165],[35,632,634,637],{"className":633},[78],[35,635,95],{"className":636,"style":111},[78,79],[35,638,640],{"className":639},[363],[35,641,643],{"className":642},[367],[35,644,646],{"className":645},[371],[35,647,650],{"className":648,"style":649},[375],"height:0.8141em;",[35,651,652,655],{"style":378},[35,653],{"className":654,"style":383},[382],[35,656,658],{"className":657},[387,388,389,390],[35,659,573],{"className":660},[78,390],[35,662,243],{"className":663},[285],[35,665],{"className":666,"style":289},[165],[35,668,578],{"className":669},[670],"minner",[35,672],{"className":673,"style":289},[165],[35,675,243],{"className":676},[285],[35,678],{"className":679,"style":289},[165],[35,681,683,686],{"className":682},[78],[35,684,95],{"className":685,"style":111},[78,79],[35,687,689],{"className":688},[363],[35,690,692],{"className":691},[367],[35,693,695],{"className":694},[371],[35,696,698],{"className":697,"style":649},[375],[35,699,700,703],{"style":378},[35,701],{"className":702,"style":383},[382],[35,704,706],{"className":705},[387,388,389,390],[35,707,709,712,716],{"className":708},[78,390],[35,710,139],{"className":711},[78,79,390],[35,713,591],{"className":714},[715,390],"mbin",[35,717,254],{"className":718},[78,390],[35,720,596],{"className":721},[296],"。",[464,724,725,728,729,757,758,786,787,831,832,918],{},[30,726,727],{},"互素的威力（Bézout 等式）","：如果 ",[35,730,732,745],{"className":731},[38],[35,733,735],{"className":734},[42],[44,736,737],{"xmlns":46},[48,738,739,743],{},[51,740,741],{},[54,742,139],{},[58,744,139],{"encoding":60},[35,746,748],{"className":747,"ariaHidden":65},[64],[35,749,751,754],{"className":750},[69],[35,752],{"className":753,"style":180},[73],[35,755,139],{"className":756},[78,79]," 和 ",[35,759,761,774],{"className":760},[38],[35,762,764],{"className":763},[42],[44,765,766],{"xmlns":46},[48,767,768,772],{},[51,769,770],{},[54,771,199],{},[58,773,199],{"encoding":60},[35,775,777],{"className":776,"ariaHidden":65},[64],[35,778,780,783],{"className":779},[69],[35,781],{"className":782,"style":211},[73],[35,784,199],{"className":785,"style":215},[78,79]," 互素，那么一定存在整数 ",[35,788,790,810],{"className":789},[38],[35,791,793],{"className":792},[42],[44,794,795],{"xmlns":46},[48,796,797,807],{},[51,798,799,802,804],{},[54,800,801],{},"u",[134,803,243],{"separator":65},[54,805,806],{},"v",[58,808,809],{"encoding":60},"u, v",[35,811,813],{"className":812,"ariaHidden":65},[64],[35,814,816,819,822,825,828],{"className":815},[69],[35,817],{"className":818,"style":107},[73],[35,820,801],{"className":821},[78,79],[35,823,243],{"className":824},[285],[35,826],{"className":827,"style":289},[165],[35,829,806],{"className":830,"style":111},[78,79]," 使得 ",[35,833,835,862],{"className":834},[38],[35,836,838],{"className":837},[42],[44,839,840],{"xmlns":46},[48,841,842,859],{},[51,843,844,846,848,851,853,855,857],{},[54,845,801],{},[54,847,139],{},[134,849,850],{},"+",[54,852,806],{},[54,854,199],{},[134,856,136],{},[252,858,254],{},[58,860,861],{"encoding":60},"un + vk = 1",[35,863,865,888,909],{"className":864,"ariaHidden":65},[64],[35,866,868,872,875,878,882,885],{"className":867},[69],[35,869],{"className":870,"style":871},[73],"height:0.6667em;vertical-align:-0.0833em;",[35,873,801],{"className":874},[78,79],[35,876,139],{"className":877},[78,79],[35,879],{"className":880,"style":881},[165],"margin-right:0.2222em;",[35,883,850],{"className":884},[715],[35,886],{"className":887,"style":881},[165],[35,889,891,894,897,900,903,906],{"className":890},[69],[35,892],{"className":893,"style":211},[73],[35,895,806],{"className":896,"style":111},[78,79],[35,898,199],{"className":899,"style":215},[78,79],[35,901],{"className":902,"style":166},[165],[35,904,136],{"className":905},[170],[35,907],{"className":908,"style":166},[165],[35,910,912,915],{"className":911},[69],[35,913],{"className":914,"style":312},[73],[35,916,254],{"className":917},[78],"。这在代数里是解题的神器。",[464,920,921,469,924],{},[30,922,923],{},"详细证明步骤",[471,925,926,2184,3895],{},[464,927,928,931,932,1014,1015,831,1056,1138,1139,1167,1168,1328,1329,931,1514,1542,1543,1571,1572,1654,1655,1921,1922,2003,2004,2084,2085,2144,2145,2183],{},[30,929,930],{},"第一步：构造一个解（存在性）。","\n因为 ",[35,933,935,964],{"className":934},[38],[35,936,938],{"className":937},[42],[44,939,940],{"xmlns":46},[48,941,942,962],{},[51,943,944,946,948,950,952,954,956,958,960],{},[54,945,231],{},[134,947,234],{},[134,949,238],{"stretchy":237},[54,951,139],{},[134,953,243],{"separator":65},[54,955,199],{},[134,957,248],{"stretchy":237},[134,959,136],{},[252,961,254],{},[58,963,257],{"encoding":60},[35,965,967,1005],{"className":966,"ariaHidden":65},[64],[35,968,970,973,978,981,984,987,990,993,996,999,1002],{"className":969},[69],[35,971],{"className":972,"style":152},[73],[35,974,976,274],{"className":975},[270],[35,977,95],{"style":273},[35,979,238],{"className":980},[278],[35,982,139],{"className":983},[78,79],[35,985,243],{"className":986},[285],[35,988],{"className":989,"style":289},[165],[35,991,199],{"className":992,"style":215},[78,79],[35,994,248],{"className":995},[296],[35,997],{"className":998,"style":166},[165],[35,1000,136],{"className":1001},[170],[35,1003],{"className":1004,"style":166},[165],[35,1006,1008,1011],{"className":1007},[69],[35,1009],{"className":1010,"style":312},[73],[35,1012,254],{"className":1013},[78],"，存在整数 ",[35,1016,1018,1035],{"className":1017},[38],[35,1019,1021],{"className":1020},[42],[44,1022,1023],{"xmlns":46},[48,1024,1025,1033],{},[51,1026,1027,1029,1031],{},[54,1028,801],{},[134,1030,243],{"separator":65},[54,1032,806],{},[58,1034,809],{"encoding":60},[35,1036,1038],{"className":1037,"ariaHidden":65},[64],[35,1039,1041,1044,1047,1050,1053],{"className":1040},[69],[35,1042],{"className":1043,"style":107},[73],[35,1045,801],{"className":1046},[78,79],[35,1048,243],{"className":1049},[285],[35,1051],{"className":1052,"style":289},[165],[35,1054,806],{"className":1055,"style":111},[78,79],[35,1057,1059,1084],{"className":1058},[38],[35,1060,1062],{"className":1061},[42],[44,1063,1064],{"xmlns":46},[48,1065,1066,1082],{},[51,1067,1068,1070,1072,1074,1076,1078,1080],{},[54,1069,801],{},[54,1071,139],{},[134,1073,850],{},[54,1075,806],{},[54,1077,199],{},[134,1079,136],{},[252,1081,254],{},[58,1083,861],{"encoding":60},[35,1085,1087,1108,1129],{"className":1086,"ariaHidden":65},[64],[35,1088,1090,1093,1096,1099,1102,1105],{"className":1089},[69],[35,1091],{"className":1092,"style":871},[73],[35,1094,801],{"className":1095},[78,79],[35,1097,139],{"className":1098},[78,79],[35,1100],{"className":1101,"style":881},[165],[35,1103,850],{"className":1104},[715],[35,1106],{"className":1107,"style":881},[165],[35,1109,1111,1114,1117,1120,1123,1126],{"className":1110},[69],[35,1112],{"className":1113,"style":211},[73],[35,1115,806],{"className":1116,"style":111},[78,79],[35,1118,199],{"className":1119,"style":215},[78,79],[35,1121],{"className":1122,"style":166},[165],[35,1124,136],{"className":1125},[170],[35,1127],{"className":1128,"style":166},[165],[35,1130,1132,1135],{"className":1131},[69],[35,1133],{"className":1134,"style":312},[73],[35,1136,254],{"className":1137},[78],"。\n我们可以把 ",[35,1140,1142,1155],{"className":1141},[38],[35,1143,1145],{"className":1144},[42],[44,1146,1147],{"xmlns":46},[48,1148,1149,1153],{},[51,1150,1151],{},[54,1152,95],{},[58,1154,95],{"encoding":60},[35,1156,1158],{"className":1157,"ariaHidden":65},[64],[35,1159,1161,1164],{"className":1160},[69],[35,1162],{"className":1163,"style":107},[73],[35,1165,95],{"className":1166,"style":111},[78,79]," 写成：\n",[35,1169,1171,1211],{"className":1170},[38],[35,1172,1174],{"className":1173},[42],[44,1175,1176],{"xmlns":46},[48,1177,1178,1208],{},[51,1179,1180,1182,1184,1190,1192],{},[54,1181,95],{},[134,1183,136],{},[330,1185,1186,1188],{},[54,1187,95],{},[252,1189,254],{},[134,1191,136],{},[330,1193,1194,1196],{},[54,1195,95],{},[51,1197,1198,1200,1202,1204,1206],{},[54,1199,801],{},[54,1201,139],{},[134,1203,850],{},[54,1205,806],{},[54,1207,199],{},[58,1209,1210],{"encoding":60},"g = g^1 = g^{un + vk}",[35,1212,1214,1232,1277],{"className":1213,"ariaHidden":65},[64],[35,1215,1217,1220,1223,1226,1229],{"className":1216},[69],[35,1218],{"className":1219,"style":107},[73],[35,1221,95],{"className":1222,"style":111},[78,79],[35,1224],{"className":1225,"style":166},[165],[35,1227,136],{"className":1228},[170],[35,1230],{"className":1231,"style":166},[165],[35,1233,1235,1239,1268,1271,1274],{"className":1234},[69],[35,1236],{"className":1237,"style":1238},[73],"height:1.0085em;vertical-align:-0.1944em;",[35,1240,1242,1245],{"className":1241},[78],[35,1243,95],{"className":1244,"style":111},[78,79],[35,1246,1248],{"className":1247},[363],[35,1249,1251],{"className":1250},[367],[35,1252,1254],{"className":1253},[371],[35,1255,1257],{"className":1256,"style":649},[375],[35,1258,1259,1262],{"style":378},[35,1260],{"className":1261,"style":383},[382],[35,1263,1265],{"className":1264},[387,388,389,390],[35,1266,254],{"className":1267},[78,390],[35,1269],{"className":1270,"style":166},[165],[35,1272,136],{"className":1273},[170],[35,1275],{"className":1276,"style":166},[165],[35,1278,1280,1284],{"className":1279},[69],[35,1281],{"className":1282,"style":1283},[73],"height:1.0435em;vertical-align:-0.1944em;",[35,1285,1287,1290],{"className":1286},[78],[35,1288,95],{"className":1289,"style":111},[78,79],[35,1291,1293],{"className":1292},[363],[35,1294,1296],{"className":1295},[367],[35,1297,1299],{"className":1298},[371],[35,1300,1302],{"className":1301,"style":353},[375],[35,1303,1304,1307],{"style":378},[35,1305],{"className":1306,"style":383},[382],[35,1308,1310],{"className":1309},[387,388,389,390],[35,1311,1313,1316,1319,1322,1325],{"className":1312},[78,390],[35,1314,801],{"className":1315},[78,79,390],[35,1317,139],{"className":1318},[78,79,390],[35,1320,850],{"className":1321},[715,390],[35,1323,806],{"className":1324,"style":111},[78,79,390],[35,1326,199],{"className":1327,"style":215},[78,79,390],"\n利用指数法则拆开：\n",[35,1330,1332,1375],{"className":1331},[38],[35,1333,1335],{"className":1334},[42],[44,1336,1337],{"xmlns":46},[48,1338,1339,1372],{},[51,1340,1341,1343,1345,1347,1353,1359,1362],{},[54,1342,95],{},[134,1344,136],{},[134,1346,238],{"stretchy":237},[330,1348,1349,1351],{},[54,1350,95],{},[54,1352,139],{},[330,1354,1355,1357],{},[134,1356,248],{"stretchy":237},[54,1358,801],{},[134,1360,1361],{},"⋅",[330,1363,1364,1366],{},[54,1365,95],{},[51,1367,1368,1370],{},[54,1369,806],{},[54,1371,199],{},[58,1373,1374],{"encoding":60},"g = (g^n)^u \\cdot g^{vk}",[35,1376,1378,1396,1473],{"className":1377,"ariaHidden":65},[64],[35,1379,1381,1384,1387,1390,1393],{"className":1380},[69],[35,1382],{"className":1383,"style":107},[73],[35,1385,95],{"className":1386,"style":111},[78,79],[35,1388],{"className":1389,"style":166},[165],[35,1391,136],{"className":1392},[170],[35,1394],{"className":1395,"style":166},[165],[35,1397,1399,1402,1405,1435,1464,1467,1470],{"className":1398},[69],[35,1400],{"className":1401,"style":152},[73],[35,1403,238],{"className":1404},[278],[35,1406,1408,1411],{"className":1407},[78],[35,1409,95],{"className":1410,"style":111},[78,79],[35,1412,1414],{"className":1413},[363],[35,1415,1417],{"className":1416},[367],[35,1418,1420],{"className":1419},[371],[35,1421,1424],{"className":1422,"style":1423},[375],"height:0.6644em;",[35,1425,1426,1429],{"style":378},[35,1427],{"className":1428,"style":383},[382],[35,1430,1432],{"className":1431},[387,388,389,390],[35,1433,139],{"className":1434},[78,79,390],[35,1436,1438,1441],{"className":1437},[296],[35,1439,248],{"className":1440},[296],[35,1442,1444],{"className":1443},[363],[35,1445,1447],{"className":1446},[367],[35,1448,1450],{"className":1449},[371],[35,1451,1453],{"className":1452,"style":1423},[375],[35,1454,1455,1458],{"style":378},[35,1456],{"className":1457,"style":383},[382],[35,1459,1461],{"className":1460},[387,388,389,390],[35,1462,801],{"className":1463},[78,79,390],[35,1465],{"className":1466,"style":881},[165],[35,1468,1361],{"className":1469},[715],[35,1471],{"className":1472,"style":881},[165],[35,1474,1476,1479],{"className":1475},[69],[35,1477],{"className":1478,"style":1283},[73],[35,1480,1482,1485],{"className":1481},[78],[35,1483,95],{"className":1484,"style":111},[78,79],[35,1486,1488],{"className":1487},[363],[35,1489,1491],{"className":1490},[367],[35,1492,1494],{"className":1493},[371],[35,1495,1497],{"className":1496,"style":353},[375],[35,1498,1499,1502],{"style":378},[35,1500],{"className":1501,"style":383},[382],[35,1503,1505],{"className":1504},[387,388,389,390],[35,1506,1508,1511],{"className":1507},[78,390],[35,1509,806],{"className":1510,"style":111},[78,79,390],[35,1512,199],{"className":1513,"style":215},[78,79,390],[35,1515,1517,1530],{"className":1516},[38],[35,1518,1520],{"className":1519},[42],[44,1521,1522],{"xmlns":46},[48,1523,1524,1528],{},[51,1525,1526],{},[54,1527,139],{},[58,1529,139],{"encoding":60},[35,1531,1533],{"className":1532,"ariaHidden":65},[64],[35,1534,1536,1539],{"className":1535},[69],[35,1537],{"className":1538,"style":180},[73],[35,1540,139],{"className":1541},[78,79]," 是 ",[35,1544,1546,1559],{"className":1545},[38],[35,1547,1549],{"className":1548},[42],[44,1550,1551],{"xmlns":46},[48,1552,1553,1557],{},[51,1554,1555],{},[54,1556,95],{},[58,1558,95],{"encoding":60},[35,1560,1562],{"className":1561,"ariaHidden":65},[64],[35,1563,1565,1568],{"className":1564},[69],[35,1566],{"className":1567,"style":107},[73],[35,1569,95],{"className":1570,"style":111},[78,79]," 的阶，所以 ",[35,1573,1575,1597],{"className":1574},[38],[35,1576,1578],{"className":1577},[42],[44,1579,1580],{"xmlns":46},[48,1581,1582,1594],{},[51,1583,1584,1590,1592],{},[330,1585,1586,1588],{},[54,1587,95],{},[54,1589,139],{},[134,1591,136],{},[54,1593,560],{},[58,1595,1596],{"encoding":60},"g^n = e",[35,1598,1600,1645],{"className":1599,"ariaHidden":65},[64],[35,1601,1603,1607,1636,1639,1642],{"className":1602},[69],[35,1604],{"className":1605,"style":1606},[73],"height:0.8588em;vertical-align:-0.1944em;",[35,1608,1610,1613],{"className":1609},[78],[35,1611,95],{"className":1612,"style":111},[78,79],[35,1614,1616],{"className":1615},[363],[35,1617,1619],{"className":1618},[367],[35,1620,1622],{"className":1621},[371],[35,1623,1625],{"className":1624,"style":1423},[375],[35,1626,1627,1630],{"style":378},[35,1628],{"className":1629,"style":383},[382],[35,1631,1633],{"className":1632},[387,388,389,390],[35,1634,139],{"className":1635},[78,79,390],[35,1637],{"className":1638,"style":166},[165],[35,1640,136],{"className":1641},[170],[35,1643],{"className":1644,"style":166},[165],[35,1646,1648,1651],{"className":1647},[69],[35,1649],{"className":1650,"style":180},[73],[35,1652,560],{"className":1653},[78,79],"（单位元）。那么：\n",[35,1656,1658,1712],{"className":1657},[38],[35,1659,1661],{"className":1660},[42],[44,1662,1663],{"xmlns":46},[48,1664,1665,1709],{},[51,1666,1667,1669,1671,1677,1679,1681,1687,1693,1695,1697,1703],{},[54,1668,95],{},[134,1670,136],{},[330,1672,1673,1675],{},[54,1674,560],{},[54,1676,801],{},[134,1678,1361],{},[134,1680,238],{"stretchy":237},[330,1682,1683,1685],{},[54,1684,95],{},[54,1686,806],{},[330,1688,1689,1691],{},[134,1690,248],{"stretchy":237},[54,1692,199],{},[134,1694,136],{},[134,1696,238],{"stretchy":237},[330,1698,1699,1701],{},[54,1700,95],{},[54,1702,806],{},[330,1704,1705,1707],{},[134,1706,248],{"stretchy":237},[54,1708,199],{},[58,1710,1711],{"encoding":60},"g = e^u \\cdot (g^v)^k = (g^v)^k",[35,1713,1715,1733,1777,1854],{"className":1714,"ariaHidden":65},[64],[35,1716,1718,1721,1724,1727,1730],{"className":1717},[69],[35,1719],{"className":1720,"style":107},[73],[35,1722,95],{"className":1723,"style":111},[78,79],[35,1725],{"className":1726,"style":166},[165],[35,1728,136],{"className":1729},[170],[35,1731],{"className":1732,"style":166},[165],[35,1734,1736,1739,1768,1771,1774],{"className":1735},[69],[35,1737],{"className":1738,"style":1423},[73],[35,1740,1742,1745],{"className":1741},[78],[35,1743,560],{"className":1744},[78,79],[35,1746,1748],{"className":1747},[363],[35,1749,1751],{"className":1750},[367],[35,1752,1754],{"className":1753},[371],[35,1755,1757],{"className":1756,"style":1423},[375],[35,1758,1759,1762],{"style":378},[35,1760],{"className":1761,"style":383},[382],[35,1763,1765],{"className":1764},[387,388,389,390],[35,1766,801],{"className":1767},[78,79,390],[35,1769],{"className":1770,"style":881},[165],[35,1772,1361],{"className":1773},[715],[35,1775],{"className":1776,"style":881},[165],[35,1778,1780,1784,1787,1816,1845,1848,1851],{"className":1779},[69],[35,1781],{"className":1782,"style":1783},[73],"height:1.0991em;vertical-align:-0.25em;",[35,1785,238],{"className":1786},[278],[35,1788,1790,1793],{"className":1789},[78],[35,1791,95],{"className":1792,"style":111},[78,79],[35,1794,1796],{"className":1795},[363],[35,1797,1799],{"className":1798},[367],[35,1800,1802],{"className":1801},[371],[35,1803,1805],{"className":1804,"style":1423},[375],[35,1806,1807,1810],{"style":378},[35,1808],{"className":1809,"style":383},[382],[35,1811,1813],{"className":1812},[387,388,389,390],[35,1814,806],{"className":1815,"style":111},[78,79,390],[35,1817,1819,1822],{"className":1818},[296],[35,1820,248],{"className":1821},[296],[35,1823,1825],{"className":1824},[363],[35,1826,1828],{"className":1827},[367],[35,1829,1831],{"className":1830},[371],[35,1832,1834],{"className":1833,"style":353},[375],[35,1835,1836,1839],{"style":378},[35,1837],{"className":1838,"style":383},[382],[35,1840,1842],{"className":1841},[387,388,389,390],[35,1843,199],{"className":1844,"style":215},[78,79,390],[35,1846],{"className":1847,"style":166},[165],[35,1849,136],{"className":1850},[170],[35,1852],{"className":1853,"style":166},[165],[35,1855,1857,1860,1863,1892],{"className":1856},[69],[35,1858],{"className":1859,"style":1783},[73],[35,1861,238],{"className":1862},[278],[35,1864,1866,1869],{"className":1865},[78],[35,1867,95],{"className":1868,"style":111},[78,79],[35,1870,1872],{"className":1871},[363],[35,1873,1875],{"className":1874},[367],[35,1876,1878],{"className":1877},[371],[35,1879,1881],{"className":1880,"style":1423},[375],[35,1882,1883,1886],{"style":378},[35,1884],{"className":1885,"style":383},[382],[35,1887,1889],{"className":1888},[387,388,389,390],[35,1890,806],{"className":1891,"style":111},[78,79,390],[35,1893,1895,1898],{"className":1894},[296],[35,1896,248],{"className":1897},[296],[35,1899,1901],{"className":1900},[363],[35,1902,1904],{"className":1903},[367],[35,1905,1907],{"className":1906},[371],[35,1908,1910],{"className":1909,"style":353},[375],[35,1911,1912,1915],{"style":378},[35,1913],{"className":1914,"style":383},[382],[35,1916,1918],{"className":1917},[387,388,389,390],[35,1919,199],{"className":1920,"style":215},[78,79,390],"\n我们找到了一个元素 ",[35,1923,1925,1947],{"className":1924},[38],[35,1926,1928],{"className":1927},[42],[44,1929,1930],{"xmlns":46},[48,1931,1932,1944],{},[51,1933,1934,1936,1938],{},[54,1935,334],{},[134,1937,136],{},[330,1939,1940,1942],{},[54,1941,95],{},[54,1943,806],{},[58,1945,1946],{"encoding":60},"x = g^v",[35,1948,1950,1968],{"className":1949,"ariaHidden":65},[64],[35,1951,1953,1956,1959,1962,1965],{"className":1952},[69],[35,1954],{"className":1955,"style":180},[73],[35,1957,334],{"className":1958},[78,79],[35,1960],{"className":1961,"style":166},[165],[35,1963,136],{"className":1964},[170],[35,1966],{"className":1967,"style":166},[165],[35,1969,1971,1974],{"className":1970},[69],[35,1972],{"className":1973,"style":1606},[73],[35,1975,1977,1980],{"className":1976},[78],[35,1978,95],{"className":1979,"style":111},[78,79],[35,1981,1983],{"className":1982},[363],[35,1984,1986],{"className":1985},[367],[35,1987,1989],{"className":1988},[371],[35,1990,1992],{"className":1991,"style":1423},[375],[35,1993,1994,1997],{"style":378},[35,1995],{"className":1996,"style":383},[382],[35,1998,2000],{"className":1999},[387,388,389,390],[35,2001,806],{"className":2002,"style":111},[78,79,390],"，它满足 ",[35,2005,2007,2028],{"className":2006},[38],[35,2008,2010],{"className":2009},[42],[44,2011,2012],{"xmlns":46},[48,2013,2014,2026],{},[51,2015,2016,2022,2024],{},[330,2017,2018,2020],{},[54,2019,334],{},[54,2021,199],{},[134,2023,136],{},[54,2025,95],{},[58,2027,343],{"encoding":60},[35,2029,2031,2075],{"className":2030,"ariaHidden":65},[64],[35,2032,2034,2037,2066,2069,2072],{"className":2033},[69],[35,2035],{"className":2036,"style":353},[73],[35,2038,2040,2043],{"className":2039},[78],[35,2041,334],{"className":2042},[78,79],[35,2044,2046],{"className":2045},[363],[35,2047,2049],{"className":2048},[367],[35,2050,2052],{"className":2051},[371],[35,2053,2055],{"className":2054,"style":353},[375],[35,2056,2057,2060],{"style":378},[35,2058],{"className":2059,"style":383},[382],[35,2061,2063],{"className":2062},[387,388,389,390],[35,2064,199],{"className":2065,"style":215},[78,79,390],[35,2067],{"className":2068,"style":166},[165],[35,2070,136],{"className":2071},[170],[35,2073],{"className":2074,"style":166},[165],[35,2076,2078,2081],{"className":2077},[69],[35,2079],{"className":2080,"style":107},[73],[35,2082,95],{"className":2083,"style":111},[78,79],"。而且 ",[35,2086,2088,2106],{"className":2087},[38],[35,2089,2091],{"className":2090},[42],[44,2092,2093],{"xmlns":46},[48,2094,2095,2103],{},[51,2096,2097],{},[330,2098,2099,2101],{},[54,2100,95],{},[54,2102,806],{},[58,2104,2105],{"encoding":60},"g^v",[35,2107,2109],{"className":2108,"ariaHidden":65},[64],[35,2110,2112,2115],{"className":2111},[69],[35,2113],{"className":2114,"style":1606},[73],[35,2116,2118,2121],{"className":2117},[78],[35,2119,95],{"className":2120,"style":111},[78,79],[35,2122,2124],{"className":2123},[363],[35,2125,2127],{"className":2126},[367],[35,2128,2130],{"className":2129},[371],[35,2131,2133],{"className":2132,"style":1423},[375],[35,2134,2135,2138],{"style":378},[35,2136],{"className":2137,"style":383},[382],[35,2139,2141],{"className":2140},[387,388,389,390],[35,2142,806],{"className":2143,"style":111},[78,79,390]," 显然在 ",[35,2146,2148,2165],{"className":2147},[38],[35,2149,2151],{"className":2150},[42],[44,2152,2153],{"xmlns":46},[48,2154,2155,2163],{},[51,2156,2157,2159,2161],{},[134,2158,427],{"stretchy":237},[54,2160,95],{},[134,2162,432],{"stretchy":237},[58,2164,435],{"encoding":60},[35,2166,2168],{"className":2167,"ariaHidden":65},[64],[35,2169,2171,2174,2177,2180],{"className":2170},[69],[35,2172],{"className":2173,"style":152},[73],[35,2175,427],{"className":2176},[278],[35,2178,95],{"className":2179,"style":111},[78,79],[35,2181,432],{"className":2182},[296]," 里。所以解是存在的。",[464,2185,2186,2189,2190,757,2268,2338,2339,2377,2378,757,2487,2592,2593,2752,2753,2959,2960,3122,3123,3151,3152,2338,3276,3314,3315,3343,3344,3372,3373,757,3401,3429,3430,3512,3513,3541,3542,3593,3594,722],{},[30,2187,2188],{},"第二步：证明只有一个解（唯一性）。","\n假设有两个解 ",[35,2191,2193,2212],{"className":2192},[38],[35,2194,2196],{"className":2195},[42],[44,2197,2198],{"xmlns":46},[48,2199,2200,2209],{},[51,2201,2202],{},[2203,2204,2205,2207],"msub",{},[54,2206,334],{},[252,2208,254],{},[58,2210,2211],{"encoding":60},"x_1",[35,2213,2215],{"className":2214,"ariaHidden":65},[64],[35,2216,2218,2222],{"className":2217},[69],[35,2219],{"className":2220,"style":2221},[73],"height:0.5806em;vertical-align:-0.15em;",[35,2223,2225,2228],{"className":2224},[78],[35,2226,334],{"className":2227},[78,79],[35,2229,2231],{"className":2230},[363],[35,2232,2235,2259],{"className":2233},[367,2234],"vlist-t2",[35,2236,2238,2254],{"className":2237},[371],[35,2239,2242],{"className":2240,"style":2241},[375],"height:0.3011em;",[35,2243,2245,2248],{"style":2244},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[35,2246],{"className":2247,"style":383},[382],[35,2249,2251],{"className":2250},[387,388,389,390],[35,2252,254],{"className":2253},[78,390],[35,2255,2258],{"className":2256},[2257],"vlist-s","​",[35,2260,2262],{"className":2261},[371],[35,2263,2266],{"className":2264,"style":2265},[375],"height:0.15em;",[35,2267],{},[35,2269,2271,2289],{"className":2270},[38],[35,2272,2274],{"className":2273},[42],[44,2275,2276],{"xmlns":46},[48,2277,2278,2286],{},[51,2279,2280],{},[2203,2281,2282,2284],{},[54,2283,334],{},[252,2285,573],{},[58,2287,2288],{"encoding":60},"x_2",[35,2290,2292],{"className":2291,"ariaHidden":65},[64],[35,2293,2295,2298],{"className":2294},[69],[35,2296],{"className":2297,"style":2221},[73],[35,2299,2301,2304],{"className":2300},[78],[35,2302,334],{"className":2303},[78,79],[35,2305,2307],{"className":2306},[363],[35,2308,2310,2330],{"className":2309},[367,2234],[35,2311,2313,2327],{"className":2312},[371],[35,2314,2316],{"className":2315,"style":2241},[375],[35,2317,2318,2321],{"style":2244},[35,2319],{"className":2320,"style":383},[382],[35,2322,2324],{"className":2323},[387,388,389,390],[35,2325,573],{"className":2326},[78,390],[35,2328,2258],{"className":2329},[2257],[35,2331,2333],{"className":2332},[371],[35,2334,2336],{"className":2335,"style":2265},[375],[35,2337],{}," 都在 ",[35,2340,2342,2359],{"className":2341},[38],[35,2343,2345],{"className":2344},[42],[44,2346,2347],{"xmlns":46},[48,2348,2349,2357],{},[51,2350,2351,2353,2355],{},[134,2352,427],{"stretchy":237},[54,2354,95],{},[134,2356,432],{"stretchy":237},[58,2358,435],{"encoding":60},[35,2360,2362],{"className":2361,"ariaHidden":65},[64],[35,2363,2365,2368,2371,2374],{"className":2364},[69],[35,2366],{"className":2367,"style":152},[73],[35,2369,427],{"className":2370},[278],[35,2372,95],{"className":2373,"style":111},[78,79],[35,2375,432],{"className":2376},[296]," 里，满足 ",[35,2379,2381,2406],{"className":2380},[38],[35,2382,2384],{"className":2383},[42],[44,2385,2386],{"xmlns":46},[48,2387,2388,2403],{},[51,2389,2390,2399,2401],{},[2391,2392,2393,2395,2397],"msubsup",{},[54,2394,334],{},[252,2396,254],{},[54,2398,199],{},[134,2400,136],{},[54,2402,95],{},[58,2404,2405],{"encoding":60},"x_1^k = g",[35,2407,2409,2478],{"className":2408,"ariaHidden":65},[64],[35,2410,2412,2416,2469,2472,2475],{"className":2411},[69],[35,2413],{"className":2414,"style":2415},[73],"height:1.0972em;vertical-align:-0.2481em;",[35,2417,2419,2422],{"className":2418},[78],[35,2420,334],{"className":2421},[78,79],[35,2423,2425],{"className":2424},[363],[35,2426,2428,2460],{"className":2427},[367,2234],[35,2429,2431,2457],{"className":2430},[371],[35,2432,2434,2446],{"className":2433,"style":353},[375],[35,2435,2437,2440],{"style":2436},"top:-2.4519em;margin-left:0em;margin-right:0.05em;",[35,2438],{"className":2439,"style":383},[382],[35,2441,2443],{"className":2442},[387,388,389,390],[35,2444,254],{"className":2445},[78,390],[35,2447,2448,2451],{"style":378},[35,2449],{"className":2450,"style":383},[382],[35,2452,2454],{"className":2453},[387,388,389,390],[35,2455,199],{"className":2456,"style":215},[78,79,390],[35,2458,2258],{"className":2459},[2257],[35,2461,2463],{"className":2462},[371],[35,2464,2467],{"className":2465,"style":2466},[375],"height:0.2481em;",[35,2468],{},[35,2470],{"className":2471,"style":166},[165],[35,2473,136],{"className":2474},[170],[35,2476],{"className":2477,"style":166},[165],[35,2479,2481,2484],{"className":2480},[69],[35,2482],{"className":2483,"style":107},[73],[35,2485,95],{"className":2486,"style":111},[78,79],[35,2488,2490,2514],{"className":2489},[38],[35,2491,2493],{"className":2492},[42],[44,2494,2495],{"xmlns":46},[48,2496,2497,2511],{},[51,2498,2499,2507,2509],{},[2391,2500,2501,2503,2505],{},[54,2502,334],{},[252,2504,573],{},[54,2506,199],{},[134,2508,136],{},[54,2510,95],{},[58,2512,2513],{"encoding":60},"x_2^k = g",[35,2515,2517,2583],{"className":2516,"ariaHidden":65},[64],[35,2518,2520,2523,2574,2577,2580],{"className":2519},[69],[35,2521],{"className":2522,"style":2415},[73],[35,2524,2526,2529],{"className":2525},[78],[35,2527,334],{"className":2528},[78,79],[35,2530,2532],{"className":2531},[363],[35,2533,2535,2566],{"className":2534},[367,2234],[35,2536,2538,2563],{"className":2537},[371],[35,2539,2541,2552],{"className":2540,"style":353},[375],[35,2542,2543,2546],{"style":2436},[35,2544],{"className":2545,"style":383},[382],[35,2547,2549],{"className":2548},[387,388,389,390],[35,2550,573],{"className":2551},[78,390],[35,2553,2554,2557],{"style":378},[35,2555],{"className":2556,"style":383},[382],[35,2558,2560],{"className":2559},[387,388,389,390],[35,2561,199],{"className":2562,"style":215},[78,79,390],[35,2564,2258],{"className":2565},[2257],[35,2567,2569],{"className":2568},[371],[35,2570,2572],{"className":2571,"style":2466},[375],[35,2573],{},[35,2575],{"className":2576,"style":166},[165],[35,2578,136],{"className":2579},[170],[35,2581],{"className":2582,"style":166},[165],[35,2584,2586,2589],{"className":2585},[69],[35,2587],{"className":2588,"style":107},[73],[35,2590,95],{"className":2591,"style":111},[78,79],"。\n那么 ",[35,2594,2596,2626],{"className":2595},[38],[35,2597,2599],{"className":2598},[42],[44,2600,2601],{"xmlns":46},[48,2602,2603,2623],{},[51,2604,2605,2613,2615],{},[2391,2606,2607,2609,2611],{},[54,2608,334],{},[252,2610,254],{},[54,2612,199],{},[134,2614,136],{},[2391,2616,2617,2619,2621],{},[54,2618,334],{},[252,2620,573],{},[54,2622,199],{},[58,2624,2625],{"encoding":60},"x_1^k = x_2^k",[35,2627,2629,2695],{"className":2628,"ariaHidden":65},[64],[35,2630,2632,2635,2686,2689,2692],{"className":2631},[69],[35,2633],{"className":2634,"style":2415},[73],[35,2636,2638,2641],{"className":2637},[78],[35,2639,334],{"className":2640},[78,79],[35,2642,2644],{"className":2643},[363],[35,2645,2647,2678],{"className":2646},[367,2234],[35,2648,2650,2675],{"className":2649},[371],[35,2651,2653,2664],{"className":2652,"style":353},[375],[35,2654,2655,2658],{"style":2436},[35,2656],{"className":2657,"style":383},[382],[35,2659,2661],{"className":2660},[387,388,389,390],[35,2662,254],{"className":2663},[78,390],[35,2665,2666,2669],{"style":378},[35,2667],{"className":2668,"style":383},[382],[35,2670,2672],{"className":2671},[387,388,389,390],[35,2673,199],{"className":2674,"style":215},[78,79,390],[35,2676,2258],{"className":2677},[2257],[35,2679,2681],{"className":2680},[371],[35,2682,2684],{"className":2683,"style":2466},[375],[35,2685],{},[35,2687],{"className":2688,"style":166},[165],[35,2690,136],{"className":2691},[170],[35,2693],{"className":2694,"style":166},[165],[35,2696,2698,2701],{"className":2697},[69],[35,2699],{"className":2700,"style":2415},[73],[35,2702,2704,2707],{"className":2703},[78],[35,2705,334],{"className":2706},[78,79],[35,2708,2710],{"className":2709},[363],[35,2711,2713,2744],{"className":2712},[367,2234],[35,2714,2716,2741],{"className":2715},[371],[35,2717,2719,2730],{"className":2718,"style":353},[375],[35,2720,2721,2724],{"style":2436},[35,2722],{"className":2723,"style":383},[382],[35,2725,2727],{"className":2726},[387,388,389,390],[35,2728,573],{"className":2729},[78,390],[35,2731,2732,2735],{"style":378},[35,2733],{"className":2734,"style":383},[382],[35,2736,2738],{"className":2737},[387,388,389,390],[35,2739,199],{"className":2740,"style":215},[78,79,390],[35,2742,2258],{"className":2743},[2257],[35,2745,2747],{"className":2746},[371],[35,2748,2750],{"className":2749,"style":2466},[375],[35,2751],{},"，也就是 ",[35,2754,2756,2798],{"className":2755},[38],[35,2757,2759],{"className":2758},[42],[44,2760,2761],{"xmlns":46},[48,2762,2763,2795],{},[51,2764,2765,2767,2773,2785,2791,2793],{},[134,2766,238],{"stretchy":237},[2203,2768,2769,2771],{},[54,2770,334],{},[252,2772,254],{},[2391,2774,2775,2777,2779],{},[54,2776,334],{},[252,2778,573],{},[51,2780,2781,2783],{},[134,2782,591],{},[252,2784,254],{},[330,2786,2787,2789],{},[134,2788,248],{"stretchy":237},[54,2790,199],{},[134,2792,136],{},[54,2794,560],{},[58,2796,2797],{"encoding":60},"(x_1 x_2^{-1})^k = e",[35,2799,2801,2950],{"className":2800,"ariaHidden":65},[64],[35,2802,2804,2808,2811,2851,2912,2941,2944,2947],{"className":2803},[69],[35,2805],{"className":2806,"style":2807},[73],"height:1.1205em;vertical-align:-0.2663em;",[35,2809,238],{"className":2810},[278],[35,2812,2814,2817],{"className":2813},[78],[35,2815,334],{"className":2816},[78,79],[35,2818,2820],{"className":2819},[363],[35,2821,2823,2843],{"className":2822},[367,2234],[35,2824,2826,2840],{"className":2825},[371],[35,2827,2829],{"className":2828,"style":2241},[375],[35,2830,2831,2834],{"style":2244},[35,2832],{"className":2833,"style":383},[382],[35,2835,2837],{"className":2836},[387,388,389,390],[35,2838,254],{"className":2839},[78,390],[35,2841,2258],{"className":2842},[2257],[35,2844,2846],{"className":2845},[371],[35,2847,2849],{"className":2848,"style":2265},[375],[35,2850],{},[35,2852,2854,2857],{"className":2853},[78],[35,2855,334],{"className":2856},[78,79],[35,2858,2860],{"className":2859},[363],[35,2861,2863,2903],{"className":2862},[367,2234],[35,2864,2866,2900],{"className":2865},[371],[35,2867,2870,2882],{"className":2868,"style":2869},[375],"height:0.8542em;",[35,2871,2873,2876],{"style":2872},"top:-2.4337em;margin-left:0em;margin-right:0.05em;",[35,2874],{"className":2875,"style":383},[382],[35,2877,2879],{"className":2878},[387,388,389,390],[35,2880,573],{"className":2881},[78,390],[35,2883,2885,2888],{"style":2884},"top:-3.1031em;margin-right:0.05em;",[35,2886],{"className":2887,"style":383},[382],[35,2889,2891],{"className":2890},[387,388,389,390],[35,2892,2894,2897],{"className":2893},[78,390],[35,2895,591],{"className":2896},[78,390],[35,2898,254],{"className":2899},[78,390],[35,2901,2258],{"className":2902},[2257],[35,2904,2906],{"className":2905},[371],[35,2907,2910],{"className":2908,"style":2909},[375],"height:0.2663em;",[35,2911],{},[35,2913,2915,2918],{"className":2914},[296],[35,2916,248],{"className":2917},[296],[35,2919,2921],{"className":2920},[363],[35,2922,2924],{"className":2923},[367],[35,2925,2927],{"className":2926},[371],[35,2928,2930],{"className":2929,"style":353},[375],[35,2931,2932,2935],{"style":378},[35,2933],{"className":2934,"style":383},[382],[35,2936,2938],{"className":2937},[387,388,389,390],[35,2939,199],{"className":2940,"style":215},[78,79,390],[35,2942],{"className":2943,"style":166},[165],[35,2945,136],{"className":2946},[170],[35,2948],{"className":2949,"style":166},[165],[35,2951,2953,2956],{"className":2952},[69],[35,2954],{"className":2955,"style":180},[73],[35,2957,560],{"className":2958},[78,79],"。\n这意味着元素 ",[35,2961,2963,2998],{"className":2962},[38],[35,2964,2966],{"className":2965},[42],[44,2967,2968],{"xmlns":46},[48,2969,2970,2995],{},[51,2971,2972,2975,2977,2983],{},[54,2973,2974],{},"y",[134,2976,136],{},[2203,2978,2979,2981],{},[54,2980,334],{},[252,2982,254],{},[2391,2984,2985,2987,2989],{},[54,2986,334],{},[252,2988,573],{},[51,2990,2991,2993],{},[134,2992,591],{},[252,2994,254],{},[58,2996,2997],{"encoding":60},"y = x_1 x_2^{-1}",[35,2999,3001,3019],{"className":3000,"ariaHidden":65},[64],[35,3002,3004,3007,3010,3013,3016],{"className":3003},[69],[35,3005],{"className":3006,"style":107},[73],[35,3008,2974],{"className":3009,"style":111},[78,79],[35,3011],{"className":3012,"style":166},[165],[35,3014,136],{"className":3015},[170],[35,3017],{"className":3018,"style":166},[165],[35,3020,3022,3025,3065],{"className":3021},[69],[35,3023],{"className":3024,"style":2807},[73],[35,3026,3028,3031],{"className":3027},[78],[35,3029,334],{"className":3030},[78,79],[35,3032,3034],{"className":3033},[363],[35,3035,3037,3057],{"className":3036},[367,2234],[35,3038,3040,3054],{"className":3039},[371],[35,3041,3043],{"className":3042,"style":2241},[375],[35,3044,3045,3048],{"style":2244},[35,3046],{"className":3047,"style":383},[382],[35,3049,3051],{"className":3050},[387,388,389,390],[35,3052,254],{"className":3053},[78,390],[35,3055,2258],{"className":3056},[2257],[35,3058,3060],{"className":3059},[371],[35,3061,3063],{"className":3062,"style":2265},[375],[35,3064],{},[35,3066,3068,3071],{"className":3067},[78],[35,3069,334],{"className":3070},[78,79],[35,3072,3074],{"className":3073},[363],[35,3075,3077,3114],{"className":3076},[367,2234],[35,3078,3080,3111],{"className":3079},[371],[35,3081,3083,3094],{"className":3082,"style":2869},[375],[35,3084,3085,3088],{"style":2872},[35,3086],{"className":3087,"style":383},[382],[35,3089,3091],{"className":3090},[387,388,389,390],[35,3092,573],{"className":3093},[78,390],[35,3095,3096,3099],{"style":2884},[35,3097],{"className":3098,"style":383},[382],[35,3100,3102],{"className":3101},[387,388,389,390],[35,3103,3105,3108],{"className":3104},[78,390],[35,3106,591],{"className":3107},[78,390],[35,3109,254],{"className":3110},[78,390],[35,3112,2258],{"className":3113},[2257],[35,3115,3117],{"className":3116},[371],[35,3118,3120],{"className":3119,"style":2909},[375],[35,3121],{}," 的阶必须整除 ",[35,3124,3126,3139],{"className":3125},[38],[35,3127,3129],{"className":3128},[42],[44,3130,3131],{"xmlns":46},[48,3132,3133,3137],{},[51,3134,3135],{},[54,3136,199],{},[58,3138,199],{"encoding":60},[35,3140,3142],{"className":3141,"ariaHidden":65},[64],[35,3143,3145,3148],{"className":3144},[69],[35,3146],{"className":3147,"style":211},[73],[35,3149,199],{"className":3150,"style":215},[78,79],"。\n同时，因为 ",[35,3153,3155,3181],{"className":3154},[38],[35,3156,3158],{"className":3157},[42],[44,3159,3160],{"xmlns":46},[48,3161,3162,3178],{},[51,3163,3164,3170,3172],{},[2203,3165,3166,3168],{},[54,3167,334],{},[252,3169,254],{},[134,3171,243],{"separator":65},[2203,3173,3174,3176],{},[54,3175,334],{},[252,3177,573],{},[58,3179,3180],{"encoding":60},"x_1, x_2",[35,3182,3184],{"className":3183,"ariaHidden":65},[64],[35,3185,3187,3190,3230,3233,3236],{"className":3186},[69],[35,3188],{"className":3189,"style":107},[73],[35,3191,3193,3196],{"className":3192},[78],[35,3194,334],{"className":3195},[78,79],[35,3197,3199],{"className":3198},[363],[35,3200,3202,3222],{"className":3201},[367,2234],[35,3203,3205,3219],{"className":3204},[371],[35,3206,3208],{"className":3207,"style":2241},[375],[35,3209,3210,3213],{"style":2244},[35,3211],{"className":3212,"style":383},[382],[35,3214,3216],{"className":3215},[387,388,389,390],[35,3217,254],{"className":3218},[78,390],[35,3220,2258],{"className":3221},[2257],[35,3223,3225],{"className":3224},[371],[35,3226,3228],{"className":3227,"style":2265},[375],[35,3229],{},[35,3231,243],{"className":3232},[285],[35,3234],{"className":3235,"style":289},[165],[35,3237,3239,3242],{"className":3238},[78],[35,3240,334],{"className":3241},[78,79],[35,3243,3245],{"className":3244},[363],[35,3246,3248,3268],{"className":3247},[367,2234],[35,3249,3251,3265],{"className":3250},[371],[35,3252,3254],{"className":3253,"style":2241},[375],[35,3255,3256,3259],{"style":2244},[35,3257],{"className":3258,"style":383},[382],[35,3260,3262],{"className":3261},[387,388,389,390],[35,3263,573],{"className":3264},[78,390],[35,3266,2258],{"className":3267},[2257],[35,3269,3271],{"className":3270},[371],[35,3272,3274],{"className":3273,"style":2265},[375],[35,3275],{},[35,3277,3279,3296],{"className":3278},[38],[35,3280,3282],{"className":3281},[42],[44,3283,3284],{"xmlns":46},[48,3285,3286,3294],{},[51,3287,3288,3290,3292],{},[134,3289,427],{"stretchy":237},[54,3291,95],{},[134,3293,432],{"stretchy":237},[58,3295,435],{"encoding":60},[35,3297,3299],{"className":3298,"ariaHidden":65},[64],[35,3300,3302,3305,3308,3311],{"className":3301},[69],[35,3303],{"className":3304,"style":152},[73],[35,3306,427],{"className":3307},[278],[35,3309,95],{"className":3310,"style":111},[78,79],[35,3312,432],{"className":3313},[296]," 里，它们的阶也都必须整除 ",[35,3316,3318,3331],{"className":3317},[38],[35,3319,3321],{"className":3320},[42],[44,3322,3323],{"xmlns":46},[48,3324,3325,3329],{},[51,3326,3327],{},[54,3328,139],{},[58,3330,139],{"encoding":60},[35,3332,3334],{"className":3333,"ariaHidden":65},[64],[35,3335,3337,3340],{"className":3336},[69],[35,3338],{"className":3339,"style":180},[73],[35,3341,139],{"className":3342},[78,79],"（群的阶）。\n所以，",[35,3345,3347,3360],{"className":3346},[38],[35,3348,3350],{"className":3349},[42],[44,3351,3352],{"xmlns":46},[48,3353,3354,3358],{},[51,3355,3356],{},[54,3357,2974],{},[58,3359,2974],{"encoding":60},[35,3361,3363],{"className":3362,"ariaHidden":65},[64],[35,3364,3366,3369],{"className":3365},[69],[35,3367],{"className":3368,"style":107},[73],[35,3370,2974],{"className":3371,"style":111},[78,79]," 的阶必须同时整除 ",[35,3374,3376,3389],{"className":3375},[38],[35,3377,3379],{"className":3378},[42],[44,3380,3381],{"xmlns":46},[48,3382,3383,3387],{},[51,3384,3385],{},[54,3386,199],{},[58,3388,199],{"encoding":60},[35,3390,3392],{"className":3391,"ariaHidden":65},[64],[35,3393,3395,3398],{"className":3394},[69],[35,3396],{"className":3397,"style":211},[73],[35,3399,199],{"className":3400,"style":215},[78,79],[35,3402,3404,3417],{"className":3403},[38],[35,3405,3407],{"className":3406},[42],[44,3408,3409],{"xmlns":46},[48,3410,3411,3415],{},[51,3412,3413],{},[54,3414,139],{},[58,3416,139],{"encoding":60},[35,3418,3420],{"className":3419,"ariaHidden":65},[64],[35,3421,3423,3426],{"className":3422},[69],[35,3424],{"className":3425,"style":180},[73],[35,3427,139],{"className":3428},[78,79],"。\n但 ",[35,3431,3433,3462],{"className":3432},[38],[35,3434,3436],{"className":3435},[42],[44,3437,3438],{"xmlns":46},[48,3439,3440,3460],{},[51,3441,3442,3444,3446,3448,3450,3452,3454,3456,3458],{},[54,3443,231],{},[134,3445,234],{},[134,3447,238],{"stretchy":237},[54,3449,139],{},[134,3451,243],{"separator":65},[54,3453,199],{},[134,3455,248],{"stretchy":237},[134,3457,136],{},[252,3459,254],{},[58,3461,257],{"encoding":60},[35,3463,3465,3503],{"className":3464,"ariaHidden":65},[64],[35,3466,3468,3471,3476,3479,3482,3485,3488,3491,3494,3497,3500],{"className":3467},[69],[35,3469],{"className":3470,"style":152},[73],[35,3472,3474,274],{"className":3473},[270],[35,3475,95],{"style":273},[35,3477,238],{"className":3478},[278],[35,3480,139],{"className":3481},[78,79],[35,3483,243],{"className":3484},[285],[35,3486],{"className":3487,"style":289},[165],[35,3489,199],{"className":3490,"style":215},[78,79],[35,3492,248],{"className":3493},[296],[35,3495],{"className":3496,"style":166},[165],[35,3498,136],{"className":3499},[170],[35,3501],{"className":3502,"style":166},[165],[35,3504,3506,3509],{"className":3505},[69],[35,3507],{"className":3508,"style":312},[73],[35,3510,254],{"className":3511},[78],"，它们没有公约数（除了1）。\n所以 ",[35,3514,3516,3529],{"className":3515},[38],[35,3517,3519],{"className":3518},[42],[44,3520,3521],{"xmlns":46},[48,3522,3523,3527],{},[51,3524,3525],{},[54,3526,2974],{},[58,3528,2974],{"encoding":60},[35,3530,3532],{"className":3531,"ariaHidden":65},[64],[35,3533,3535,3538],{"className":3534},[69],[35,3536],{"className":3537,"style":107},[73],[35,3539,2974],{"className":3540,"style":111},[78,79]," 的阶只能是 1，即 ",[35,3543,3545,3563],{"className":3544},[38],[35,3546,3548],{"className":3547},[42],[44,3549,3550],{"xmlns":46},[48,3551,3552,3560],{},[51,3553,3554,3556,3558],{},[54,3555,2974],{},[134,3557,136],{},[54,3559,560],{},[58,3561,3562],{"encoding":60},"y = e",[35,3564,3566,3584],{"className":3565,"ariaHidden":65},[64],[35,3567,3569,3572,3575,3578,3581],{"className":3568},[69],[35,3570],{"className":3571,"style":107},[73],[35,3573,2974],{"className":3574,"style":111},[78,79],[35,3576],{"className":3577,"style":166},[165],[35,3579,136],{"className":3580},[170],[35,3582],{"className":3583,"style":166},[165],[35,3585,3587,3590],{"className":3586},[69],[35,3588],{"className":3589,"style":180},[73],[35,3591,560],{"className":3592},[78,79],"。\n推导出 ",[35,3595,3597,3654],{"className":3596},[38],[35,3598,3600],{"className":3599},[42],[44,3601,3602],{"xmlns":46},[48,3603,3604,3651],{},[51,3605,3606,3612,3624,3626,3628,3632,3635,3637,3643,3645],{},[2203,3607,3608,3610],{},[54,3609,334],{},[252,3611,254],{},[2391,3613,3614,3616,3618],{},[54,3615,334],{},[252,3617,573],{},[51,3619,3620,3622],{},[134,3621,591],{},[252,3623,254],{},[134,3625,136],{},[54,3627,560],{},[3629,3630,3631],"mtext",{},"  ",[134,3633,3634],{},"⟹",[3629,3636,3631],{},[2203,3638,3639,3641],{},[54,3640,334],{},[252,3642,254],{},[134,3644,136],{},[2203,3646,3647,3649],{},[54,3648,334],{},[252,3650,573],{},[58,3652,3653],{"encoding":60},"x_1 x_2^{-1} = e \\implies x_1 = x_2",[35,3655,3657,3769,3794,3849],{"className":3656,"ariaHidden":65},[64],[35,3658,3660,3663,3703,3760,3763,3766],{"className":3659},[69],[35,3661],{"className":3662,"style":2807},[73],[35,3664,3666,3669],{"className":3665},[78],[35,3667,334],{"className":3668},[78,79],[35,3670,3672],{"className":3671},[363],[35,3673,3675,3695],{"className":3674},[367,2234],[35,3676,3678,3692],{"className":3677},[371],[35,3679,3681],{"className":3680,"style":2241},[375],[35,3682,3683,3686],{"style":2244},[35,3684],{"className":3685,"style":383},[382],[35,3687,3689],{"className":3688},[387,388,389,390],[35,3690,254],{"className":3691},[78,390],[35,3693,2258],{"className":3694},[2257],[35,3696,3698],{"className":3697},[371],[35,3699,3701],{"className":3700,"style":2265},[375],[35,3702],{},[35,3704,3706,3709],{"className":3705},[78],[35,3707,334],{"className":3708},[78,79],[35,3710,3712],{"className":3711},[363],[35,3713,3715,3752],{"className":3714},[367,2234],[35,3716,3718,3749],{"className":3717},[371],[35,3719,3721,3732],{"className":3720,"style":2869},[375],[35,3722,3723,3726],{"style":2872},[35,3724],{"className":3725,"style":383},[382],[35,3727,3729],{"className":3728},[387,388,389,390],[35,3730,573],{"className":3731},[78,390],[35,3733,3734,3737],{"style":2884},[35,3735],{"className":3736,"style":383},[382],[35,3738,3740],{"className":3739},[387,388,389,390],[35,3741,3743,3746],{"className":3742},[78,390],[35,3744,591],{"className":3745},[78,390],[35,3747,254],{"className":3748},[78,390],[35,3750,2258],{"className":3751},[2257],[35,3753,3755],{"className":3754},[371],[35,3756,3758],{"className":3757,"style":2909},[375],[35,3759],{},[35,3761],{"className":3762,"style":166},[165],[35,3764,136],{"className":3765},[170],[35,3767],{"className":3768,"style":166},[165],[35,3770,3772,3776,3779,3782,3785,3788,3791],{"className":3771},[69],[35,3773],{"className":3774,"style":3775},[73],"height:0.549em;vertical-align:-0.024em;",[35,3777,560],{"className":3778},[78,79],[35,3780],{"className":3781,"style":166},[165],[35,3783],{"className":3784,"style":166},[165],[35,3786,3634],{"className":3787},[170],[35,3789],{"className":3790,"style":166},[165],[35,3792],{"className":3793,"style":166},[165],[35,3795,3797,3800,3840,3843,3846],{"className":3796},[69],[35,3798],{"className":3799,"style":2221},[73],[35,3801,3803,3806],{"className":3802},[78],[35,3804,334],{"className":3805},[78,79],[35,3807,3809],{"className":3808},[363],[35,3810,3812,3832],{"className":3811},[367,2234],[35,3813,3815,3829],{"className":3814},[371],[35,3816,3818],{"className":3817,"style":2241},[375],[35,3819,3820,3823],{"style":2244},[35,3821],{"className":3822,"style":383},[382],[35,3824,3826],{"className":3825},[387,388,389,390],[35,3827,254],{"className":3828},[78,390],[35,3830,2258],{"className":3831},[2257],[35,3833,3835],{"className":3834},[371],[35,3836,3838],{"className":3837,"style":2265},[375],[35,3839],{},[35,3841],{"className":3842,"style":166},[165],[35,3844,136],{"className":3845},[170],[35,3847],{"className":3848,"style":166},[165],[35,3850,3852,3855],{"className":3851},[69],[35,3853],{"className":3854,"style":2221},[73],[35,3856,3858,3861],{"className":3857},[78],[35,3859,334],{"className":3860},[78,79],[35,3862,3864],{"className":3863},[363],[35,3865,3867,3887],{"className":3866},[367,2234],[35,3868,3870,3884],{"className":3869},[371],[35,3871,3873],{"className":3872,"style":2241},[375],[35,3874,3875,3878],{"style":2244},[35,3876],{"className":3877,"style":383},[382],[35,3879,3881],{"className":3880},[387,388,389,390],[35,3882,573],{"className":3883},[78,390],[35,3885,2258],{"className":3886},[2257],[35,3888,3890],{"className":3889},[371],[35,3891,3893],{"className":3892,"style":2265},[375],[35,3894],{},[464,3896,3897,3900],{},[30,3898,3899],{},"结论","：解存在且唯一。",[14,3902],{},[22,3904,3906],{"id":3905},"题目-3有理数加法群的子群","题目 3：有理数加法群的子群",[27,3908,3909,3911,3912,3967],{},[30,3910,32],{}," 证明有理数加法群 ",[35,3913,3915,3939],{"className":3914},[38],[35,3916,3918],{"className":3917},[42],[44,3919,3920],{"xmlns":46},[48,3921,3922,3936],{},[51,3923,3924,3926,3930,3932,3934],{},[134,3925,238],{"stretchy":237},[54,3927,3929],{"mathvariant":3928},"double-struck","Q",[134,3931,243],{"separator":65},[134,3933,850],{},[134,3935,248],{"stretchy":237},[58,3937,3938],{"encoding":60},"(\\mathbb{Q}, +)",[35,3940,3942],{"className":3941,"ariaHidden":65},[64],[35,3943,3945,3948,3951,3955,3958,3961,3964],{"className":3944},[69],[35,3946],{"className":3947,"style":152},[73],[35,3949,238],{"className":3950},[278],[35,3952,3929],{"className":3953},[78,3954],"mathbb",[35,3956,243],{"className":3957},[285],[35,3959],{"className":3960,"style":289},[165],[35,3962,850],{"className":3963},[78],[35,3965,248],{"className":3966},[296]," 的任一有限生成子群都是循环群。",[27,3969,3970],{},[30,3971,459],{},[461,3973,3974,3980],{},[464,3975,3976,3979],{},[30,3977,3978],{},"直观理解","：有理数很稠密，但如果你只取“有限个”有理数通过加减法生成一个群，它们其实都“卡”在一个共同分母的网格上。",[464,3981,3982,3985,3986],{},[30,3983,3984],{},"详细讲解","：\n",[471,3987,3988,4601,4783,5188,5915,6190,6334,6340],{},[464,3989,3990,3991,4021,4022,722],{},"设这个子群 ",[35,3992,3994,4008],{"className":3993},[38],[35,3995,3997],{"className":3996},[42],[44,3998,3999],{"xmlns":46},[48,4000,4001,4006],{},[51,4002,4003],{},[54,4004,4005],{},"H",[58,4007,4005],{"encoding":60},[35,4009,4011],{"className":4010,"ariaHidden":65},[64],[35,4012,4014,4017],{"className":4013},[69],[35,4015],{"className":4016,"style":74},[73],[35,4018,4005],{"className":4019,"style":4020},[78,79],"margin-right:0.0813em;"," 是由有限个有理数生成的：",[35,4023,4025,4099],{"className":4024},[38],[35,4026,4028],{"className":4027},[42],[44,4029,4030],{"xmlns":46},[48,4031,4032,4096],{},[51,4033,4034,4036,4038,4040,4057,4059,4073,4075,4077,4079,4094],{},[54,4035,4005],{},[134,4037,136],{},[134,4039,427],{"stretchy":237},[4041,4042,4043,4050],"mfrac",{},[2203,4044,4045,4048],{},[54,4046,4047],{},"r",[252,4049,254],{},[2203,4051,4052,4055],{},[54,4053,4054],{},"s",[252,4056,254],{},[134,4058,243],{"separator":65},[4041,4060,4061,4067],{},[2203,4062,4063,4065],{},[54,4064,4047],{},[252,4066,573],{},[2203,4068,4069,4071],{},[54,4070,4054],{},[252,4072,573],{},[134,4074,243],{"separator":65},[134,4076,578],{},[134,4078,243],{"separator":65},[4041,4080,4081,4088],{},[2203,4082,4083,4085],{},[54,4084,4047],{},[54,4086,4087],{},"m",[2203,4089,4090,4092],{},[54,4091,4054],{},[54,4093,4087],{},[134,4095,432],{"stretchy":237},[58,4097,4098],{"encoding":60},"H = \\langle \\frac{r_1}{s_1}, \\frac{r_2}{s_2}, \\dots, \\frac{r_m}{s_m} \\rangle",[35,4100,4102,4120],{"className":4101,"ariaHidden":65},[64],[35,4103,4105,4108,4111,4114,4117],{"className":4104},[69],[35,4106],{"className":4107,"style":74},[73],[35,4109,4005],{"className":4110,"style":4020},[78,79],[35,4112],{"className":4113,"style":166},[165],[35,4115,136],{"className":4116},[170],[35,4118],{"className":4119,"style":166},[165],[35,4121,4123,4127,4130,4289,4292,4295,4437,4440,4443,4446,4449,4452,4455,4598],{"className":4122},[69],[35,4124],{"className":4125,"style":4126},[73],"height:1.1951em;vertical-align:-0.4451em;",[35,4128,427],{"className":4129},[278],[35,4131,4133,4137,4286],{"className":4132},[78],[35,4134],{"className":4135},[278,4136],"nulldelimiter",[35,4138,4140],{"className":4139},[4041],[35,4141,4143,4277],{"className":4142},[367,2234],[35,4144,4146,4274],{"className":4145},[371],[35,4147,4150,4209,4220],{"className":4148,"style":4149},[375],"height:0.7115em;",[35,4151,4153,4157],{"style":4152},"top:-2.655em;",[35,4154],{"className":4155,"style":4156},[382],"height:3em;",[35,4158,4160],{"className":4159},[387,388,389,390],[35,4161,4163],{"className":4162},[78,390],[35,4164,4166,4169],{"className":4165},[78,390],[35,4167,4054],{"className":4168},[78,79,390],[35,4170,4172],{"className":4171},[363],[35,4173,4175,4200],{"className":4174},[367,2234],[35,4176,4178,4197],{"className":4177},[371],[35,4179,4182],{"className":4180,"style":4181},[375],"height:0.3173em;",[35,4183,4185,4189],{"style":4184},"top:-2.357em;margin-left:0em;margin-right:0.0714em;",[35,4186],{"className":4187,"style":4188},[382],"height:2.5em;",[35,4190,4194],{"className":4191},[387,4192,4193,390],"reset-size3","size1",[35,4195,254],{"className":4196},[78,390],[35,4198,2258],{"className":4199},[2257],[35,4201,4203],{"className":4202},[371],[35,4204,4207],{"className":4205,"style":4206},[375],"height:0.143em;",[35,4208],{},[35,4210,4212,4215],{"style":4211},"top:-3.23em;",[35,4213],{"className":4214,"style":4156},[382],[35,4216],{"className":4217,"style":4219},[4218],"frac-line","border-bottom-width:0.04em;",[35,4221,4223,4226],{"style":4222},"top:-3.4101em;",[35,4224],{"className":4225,"style":4156},[382],[35,4227,4229],{"className":4228},[387,388,389,390],[35,4230,4232],{"className":4231},[78,390],[35,4233,4235,4239],{"className":4234},[78,390],[35,4236,4047],{"className":4237,"style":4238},[78,79,390],"margin-right:0.0278em;",[35,4240,4242],{"className":4241},[363],[35,4243,4245,4266],{"className":4244},[367,2234],[35,4246,4248,4263],{"className":4247},[371],[35,4249,4251],{"className":4250,"style":4181},[375],[35,4252,4254,4257],{"style":4253},"top:-2.357em;margin-left:-0.0278em;margin-right:0.0714em;",[35,4255],{"className":4256,"style":4188},[382],[35,4258,4260],{"className":4259},[387,4192,4193,390],[35,4261,254],{"className":4262},[78,390],[35,4264,2258],{"className":4265},[2257],[35,4267,4269],{"className":4268},[371],[35,4270,4272],{"className":4271,"style":4206},[375],[35,4273],{},[35,4275,2258],{"className":4276},[2257],[35,4278,4280],{"className":4279},[371],[35,4281,4284],{"className":4282,"style":4283},[375],"height:0.4451em;",[35,4285],{},[35,4287],{"className":4288},[296,4136],[35,4290,243],{"className":4291},[285],[35,4293],{"className":4294,"style":289},[165],[35,4296,4298,4301,4434],{"className":4297},[78],[35,4299],{"className":4300},[278,4136],[35,4302,4304],{"className":4303},[4041],[35,4305,4307,4426],{"className":4306},[367,2234],[35,4308,4310,4423],{"className":4309},[371],[35,4311,4313,4364,4372],{"className":4312,"style":4149},[375],[35,4314,4315,4318],{"style":4152},[35,4316],{"className":4317,"style":4156},[382],[35,4319,4321],{"className":4320},[387,388,389,390],[35,4322,4324],{"className":4323},[78,390],[35,4325,4327,4330],{"className":4326},[78,390],[35,4328,4054],{"className":4329},[78,79,390],[35,4331,4333],{"className":4332},[363],[35,4334,4336,4356],{"className":4335},[367,2234],[35,4337,4339,4353],{"className":4338},[371],[35,4340,4342],{"className":4341,"style":4181},[375],[35,4343,4344,4347],{"style":4184},[35,4345],{"className":4346,"style":4188},[382],[35,4348,4350],{"className":4349},[387,4192,4193,390],[35,4351,573],{"className":4352},[78,390],[35,4354,2258],{"className":4355},[2257],[35,4357,4359],{"className":4358},[371],[35,4360,4362],{"className":4361,"style":4206},[375],[35,4363],{},[35,4365,4366,4369],{"style":4211},[35,4367],{"className":4368,"style":4156},[382],[35,4370],{"className":4371,"style":4219},[4218],[35,4373,4374,4377],{"style":4222},[35,4375],{"className":4376,"style":4156},[382],[35,4378,4380],{"className":4379},[387,388,389,390],[35,4381,4383],{"className":4382},[78,390],[35,4384,4386,4389],{"className":4385},[78,390],[35,4387,4047],{"className":4388,"style":4238},[78,79,390],[35,4390,4392],{"className":4391},[363],[35,4393,4395,4415],{"className":4394},[367,2234],[35,4396,4398,4412],{"className":4397},[371],[35,4399,4401],{"className":4400,"style":4181},[375],[35,4402,4403,4406],{"style":4253},[35,4404],{"className":4405,"style":4188},[382],[35,4407,4409],{"className":4408},[387,4192,4193,390],[35,4410,573],{"className":4411},[78,390],[35,4413,2258],{"className":4414},[2257],[35,4416,4418],{"className":4417},[371],[35,4419,4421],{"className":4420,"style":4206},[375],[35,4422],{},[35,4424,2258],{"className":4425},[2257],[35,4427,4429],{"className":4428},[371],[35,4430,4432],{"className":4431,"style":4283},[375],[35,4433],{},[35,4435],{"className":4436},[296,4136],[35,4438,243],{"className":4439},[285],[35,4441],{"className":4442,"style":289},[165],[35,4444,578],{"className":4445},[670],[35,4447],{"className":4448,"style":289},[165],[35,4450,243],{"className":4451},[285],[35,4453],{"className":4454,"style":289},[165],[35,4456,4458,4461,4595],{"className":4457},[78],[35,4459],{"className":4460},[278,4136],[35,4462,4464],{"className":4463},[4041],[35,4465,4467,4587],{"className":4466},[367,2234],[35,4468,4470,4584],{"className":4469},[371],[35,4471,4473,4525,4533],{"className":4472,"style":4149},[375],[35,4474,4475,4478],{"style":4152},[35,4476],{"className":4477,"style":4156},[382],[35,4479,4481],{"className":4480},[387,388,389,390],[35,4482,4484],{"className":4483},[78,390],[35,4485,4487,4490],{"className":4486},[78,390],[35,4488,4054],{"className":4489},[78,79,390],[35,4491,4493],{"className":4492},[363],[35,4494,4496,4517],{"className":4495},[367,2234],[35,4497,4499,4514],{"className":4498},[371],[35,4500,4503],{"className":4501,"style":4502},[375],"height:0.1645em;",[35,4504,4505,4508],{"style":4184},[35,4506],{"className":4507,"style":4188},[382],[35,4509,4511],{"className":4510},[387,4192,4193,390],[35,4512,4087],{"className":4513},[78,79,390],[35,4515,2258],{"className":4516},[2257],[35,4518,4520],{"className":4519},[371],[35,4521,4523],{"className":4522,"style":4206},[375],[35,4524],{},[35,4526,4527,4530],{"style":4211},[35,4528],{"className":4529,"style":4156},[382],[35,4531],{"className":4532,"style":4219},[4218],[35,4534,4535,4538],{"style":4222},[35,4536],{"className":4537,"style":4156},[382],[35,4539,4541],{"className":4540},[387,388,389,390],[35,4542,4544],{"className":4543},[78,390],[35,4545,4547,4550],{"className":4546},[78,390],[35,4548,4047],{"className":4549,"style":4238},[78,79,390],[35,4551,4553],{"className":4552},[363],[35,4554,4556,4576],{"className":4555},[367,2234],[35,4557,4559,4573],{"className":4558},[371],[35,4560,4562],{"className":4561,"style":4502},[375],[35,4563,4564,4567],{"style":4253},[35,4565],{"className":4566,"style":4188},[382],[35,4568,4570],{"className":4569},[387,4192,4193,390],[35,4571,4087],{"className":4572},[78,79,390],[35,4574,2258],{"className":4575},[2257],[35,4577,4579],{"className":4578},[371],[35,4580,4582],{"className":4581,"style":4206},[375],[35,4583],{},[35,4585,2258],{"className":4586},[2257],[35,4588,4590],{"className":4589},[371],[35,4591,4593],{"className":4592,"style":4283},[375],[35,4594],{},[35,4596],{"className":4597},[296,4136],[35,4599,432],{"className":4600},[296],[464,4602,4603,4606,4607,4637,4638,4779,4780,722],{},[30,4604,4605],{},"通分","：令 ",[35,4608,4610,4624],{"className":4609},[38],[35,4611,4613],{"className":4612},[42],[44,4614,4615],{"xmlns":46},[48,4616,4617,4622],{},[51,4618,4619],{},[54,4620,4621],{},"S",[58,4623,4621],{"encoding":60},[35,4625,4627],{"className":4626,"ariaHidden":65},[64],[35,4628,4630,4633],{"className":4629},[69],[35,4631],{"className":4632,"style":74},[73],[35,4634,4621],{"className":4635,"style":4636},[78,79],"margin-right:0.0576em;"," 为所有分母 ",[35,4639,4641,4671],{"className":4640},[38],[35,4642,4644],{"className":4643},[42],[44,4645,4646],{"xmlns":46},[48,4647,4648,4668],{},[51,4649,4650,4656,4658,4660,4662],{},[2203,4651,4652,4654],{},[54,4653,4054],{},[252,4655,254],{},[134,4657,243],{"separator":65},[134,4659,578],{},[134,4661,243],{"separator":65},[2203,4663,4664,4666],{},[54,4665,4054],{},[54,4667,4087],{},[58,4669,4670],{"encoding":60},"s_1, \\dots, s_m",[35,4672,4674],{"className":4673,"ariaHidden":65},[64],[35,4675,4677,4680,4720,4723,4726,4729,4732,4735,4738],{"className":4676},[69],[35,4678],{"className":4679,"style":107},[73],[35,4681,4683,4686],{"className":4682},[78],[35,4684,4054],{"className":4685},[78,79],[35,4687,4689],{"className":4688},[363],[35,4690,4692,4712],{"className":4691},[367,2234],[35,4693,4695,4709],{"className":4694},[371],[35,4696,4698],{"className":4697,"style":2241},[375],[35,4699,4700,4703],{"style":2244},[35,4701],{"className":4702,"style":383},[382],[35,4704,4706],{"className":4705},[387,388,389,390],[35,4707,254],{"className":4708},[78,390],[35,4710,2258],{"className":4711},[2257],[35,4713,4715],{"className":4714},[371],[35,4716,4718],{"className":4717,"style":2265},[375],[35,4719],{},[35,4721,243],{"className":4722},[285],[35,4724],{"className":4725,"style":289},[165],[35,4727,578],{"className":4728},[670],[35,4730],{"className":4731,"style":289},[165],[35,4733,243],{"className":4734},[285],[35,4736],{"className":4737,"style":289},[165],[35,4739,4741,4744],{"className":4740},[78],[35,4742,4054],{"className":4743},[78,79],[35,4745,4747],{"className":4746},[363],[35,4748,4750,4771],{"className":4749},[367,2234],[35,4751,4753,4768],{"className":4752},[371],[35,4754,4757],{"className":4755,"style":4756},[375],"height:0.1514em;",[35,4758,4759,4762],{"style":2244},[35,4760],{"className":4761,"style":383},[382],[35,4763,4765],{"className":4764},[387,388,389,390],[35,4766,4087],{"className":4767},[78,79,390],[35,4769,2258],{"className":4770},[2257],[35,4772,4774],{"className":4773},[371],[35,4775,4777],{"className":4776,"style":2265},[375],[35,4778],{}," 的",[30,4781,4782],{},"最小公倍数",[464,4784,4785,4786,4969,4970,5113,5114,5187],{},"那么，每一个生成元 ",[35,4787,4789,4816],{"className":4788},[38],[35,4790,4792],{"className":4791},[42],[44,4793,4794],{"xmlns":46},[48,4795,4796,4813],{},[51,4797,4798],{},[4041,4799,4800,4807],{},[2203,4801,4802,4804],{},[54,4803,4047],{},[54,4805,4806],{},"i",[2203,4808,4809,4811],{},[54,4810,4054],{},[54,4812,4806],{},[58,4814,4815],{"encoding":60},"\\frac{r_i}{s_i}",[35,4817,4819],{"className":4818,"ariaHidden":65},[64],[35,4820,4822,4826],{"className":4821},[69],[35,4823],{"className":4824,"style":4825},[73],"height:1.1566em;vertical-align:-0.4451em;",[35,4827,4829,4832,4966],{"className":4828},[78],[35,4830],{"className":4831},[278,4136],[35,4833,4835],{"className":4834},[4041],[35,4836,4838,4958],{"className":4837},[367,2234],[35,4839,4841,4955],{"className":4840},[371],[35,4842,4844,4896,4904],{"className":4843,"style":4149},[375],[35,4845,4846,4849],{"style":4152},[35,4847],{"className":4848,"style":4156},[382],[35,4850,4852],{"className":4851},[387,388,389,390],[35,4853,4855],{"className":4854},[78,390],[35,4856,4858,4861],{"className":4857},[78,390],[35,4859,4054],{"className":4860},[78,79,390],[35,4862,4864],{"className":4863},[363],[35,4865,4867,4888],{"className":4866},[367,2234],[35,4868,4870,4885],{"className":4869},[371],[35,4871,4874],{"className":4872,"style":4873},[375],"height:0.3281em;",[35,4875,4876,4879],{"style":4184},[35,4877],{"className":4878,"style":4188},[382],[35,4880,4882],{"className":4881},[387,4192,4193,390],[35,4883,4806],{"className":4884},[78,79,390],[35,4886,2258],{"className":4887},[2257],[35,4889,4891],{"className":4890},[371],[35,4892,4894],{"className":4893,"style":4206},[375],[35,4895],{},[35,4897,4898,4901],{"style":4211},[35,4899],{"className":4900,"style":4156},[382],[35,4902],{"className":4903,"style":4219},[4218],[35,4905,4906,4909],{"style":4222},[35,4907],{"className":4908,"style":4156},[382],[35,4910,4912],{"className":4911},[387,388,389,390],[35,4913,4915],{"className":4914},[78,390],[35,4916,4918,4921],{"className":4917},[78,390],[35,4919,4047],{"className":4920,"style":4238},[78,79,390],[35,4922,4924],{"className":4923},[363],[35,4925,4927,4947],{"className":4926},[367,2234],[35,4928,4930,4944],{"className":4929},[371],[35,4931,4933],{"className":4932,"style":4873},[375],[35,4934,4935,4938],{"style":4253},[35,4936],{"className":4937,"style":4188},[382],[35,4939,4941],{"className":4940},[387,4192,4193,390],[35,4942,4806],{"className":4943},[78,79,390],[35,4945,2258],{"className":4946},[2257],[35,4948,4950],{"className":4949},[371],[35,4951,4953],{"className":4952,"style":4206},[375],[35,4954],{},[35,4956,2258],{"className":4957},[2257],[35,4959,4961],{"className":4960},[371],[35,4962,4964],{"className":4963,"style":4283},[375],[35,4965],{},[35,4967],{"className":4968},[296,4136]," 都可以写成 ",[35,4971,4973,4995],{"className":4972},[38],[35,4974,4976],{"className":4975},[42],[44,4977,4978],{"xmlns":46},[48,4979,4980,4992],{},[51,4981,4982],{},[4041,4983,4984,4990],{},[2203,4985,4986,4988],{},[54,4987,199],{},[54,4989,4806],{},[54,4991,4621],{},[58,4993,4994],{"encoding":60},"\\frac{k_i}{S}",[35,4996,4998],{"className":4997,"ariaHidden":65},[64],[35,4999,5001,5005],{"className":5000},[69],[35,5002],{"className":5003,"style":5004},[73],"height:1.2412em;vertical-align:-0.345em;",[35,5006,5008,5011,5110],{"className":5007},[78],[35,5009],{"className":5010},[278,4136],[35,5012,5014],{"className":5013},[4041],[35,5015,5017,5101],{"className":5016},[367,2234],[35,5018,5020,5098],{"className":5019},[371],[35,5021,5024,5038,5046],{"className":5022,"style":5023},[375],"height:0.8962em;",[35,5025,5026,5029],{"style":4152},[35,5027],{"className":5028,"style":4156},[382],[35,5030,5032],{"className":5031},[387,388,389,390],[35,5033,5035],{"className":5034},[78,390],[35,5036,4621],{"className":5037,"style":4636},[78,79,390],[35,5039,5040,5043],{"style":4211},[35,5041],{"className":5042,"style":4156},[382],[35,5044],{"className":5045,"style":4219},[4218],[35,5047,5048,5051],{"style":4222},[35,5049],{"className":5050,"style":4156},[382],[35,5052,5054],{"className":5053},[387,388,389,390],[35,5055,5057],{"className":5056},[78,390],[35,5058,5060,5063],{"className":5059},[78,390],[35,5061,199],{"className":5062,"style":215},[78,79,390],[35,5064,5066],{"className":5065},[363],[35,5067,5069,5090],{"className":5068},[367,2234],[35,5070,5072,5087],{"className":5071},[371],[35,5073,5075],{"className":5074,"style":4873},[375],[35,5076,5078,5081],{"style":5077},"top:-2.357em;margin-left:-0.0315em;margin-right:0.0714em;",[35,5079],{"className":5080,"style":4188},[382],[35,5082,5084],{"className":5083},[387,4192,4193,390],[35,5085,4806],{"className":5086},[78,79,390],[35,5088,2258],{"className":5089},[2257],[35,5091,5093],{"className":5092},[371],[35,5094,5096],{"className":5095,"style":4206},[375],[35,5097],{},[35,5099,2258],{"className":5100},[2257],[35,5102,5104],{"className":5103},[371],[35,5105,5108],{"className":5106,"style":5107},[375],"height:0.345em;",[35,5109],{},[35,5111],{"className":5112},[296,4136]," 的形式（其中 ",[35,5115,5117,5135],{"className":5116},[38],[35,5118,5120],{"className":5119},[42],[44,5121,5122],{"xmlns":46},[48,5123,5124,5132],{},[51,5125,5126],{},[2203,5127,5128,5130],{},[54,5129,199],{},[54,5131,4806],{},[58,5133,5134],{"encoding":60},"k_i",[35,5136,5138],{"className":5137,"ariaHidden":65},[64],[35,5139,5141,5145],{"className":5140},[69],[35,5142],{"className":5143,"style":5144},[73],"height:0.8444em;vertical-align:-0.15em;",[35,5146,5148,5151],{"className":5147},[78],[35,5149,199],{"className":5150,"style":215},[78,79],[35,5152,5154],{"className":5153},[363],[35,5155,5157,5179],{"className":5156},[367,2234],[35,5158,5160,5176],{"className":5159},[371],[35,5161,5164],{"className":5162,"style":5163},[375],"height:0.3117em;",[35,5165,5167,5170],{"style":5166},"top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;",[35,5168],{"className":5169,"style":383},[382],[35,5171,5173],{"className":5172},[387,388,389,390],[35,5174,4806],{"className":5175},[78,79,390],[35,5177,2258],{"className":5178},[2257],[35,5180,5182],{"className":5181},[371],[35,5183,5185],{"className":5184,"style":2265},[375],[35,5186],{}," 是整数）。",[464,5189,5190,5191,5219,5220],{},"这意味着，群 ",[35,5192,5194,5207],{"className":5193},[38],[35,5195,5197],{"className":5196},[42],[44,5198,5199],{"xmlns":46},[48,5200,5201,5205],{},[51,5202,5203],{},[54,5204,4005],{},[58,5206,4005],{"encoding":60},[35,5208,5210],{"className":5209,"ariaHidden":65},[64],[35,5211,5213,5216],{"className":5212},[69],[35,5214],{"className":5215,"style":74},[73],[35,5217,4005],{"className":5218,"style":4020},[78,79]," 中的任何元素都可以写成：\n",[35,5221,5223,5316],{"className":5222},[38],[35,5224,5226],{"className":5225},[42],[44,5227,5228],{"xmlns":46},[48,5229,5230,5313],{},[51,5231,5232,5234,5236,5242,5252,5254,5257,5259,5265,5275,5277],{},[54,5233,334],{},[134,5235,136],{},[2203,5237,5238,5240],{},[54,5239,139],{},[252,5241,254],{},[4041,5243,5244,5250],{},[2203,5245,5246,5248],{},[54,5247,199],{},[252,5249,254],{},[54,5251,4621],{},[134,5253,850],{},[134,5255,5256],{},"⋯",[134,5258,850],{},[2203,5260,5261,5263],{},[54,5262,139],{},[54,5264,4087],{},[4041,5266,5267,5273],{},[2203,5268,5269,5271],{},[54,5270,199],{},[54,5272,4087],{},[54,5274,4621],{},[134,5276,136],{},[4041,5278,5279,5311],{},[51,5280,5281,5287,5293,5295,5297,5299,5305],{},[2203,5282,5283,5285],{},[54,5284,139],{},[252,5286,254],{},[2203,5288,5289,5291],{},[54,5290,199],{},[252,5292,254],{},[134,5294,850],{},[134,5296,5256],{},[134,5298,850],{},[2203,5300,5301,5303],{},[54,5302,139],{},[54,5304,4087],{},[2203,5306,5307,5309],{},[54,5308,199],{},[54,5310,4087],{},[54,5312,4621],{},[58,5314,5315],{"encoding":60},"x = n_1 \\frac{k_1}{S} + \\dots + n_m \\frac{k_m}{S} = \\frac{n_1 k_1 + \\dots + n_m k_m}{S}",[35,5317,5319,5337,5497,5515,5675],{"className":5318,"ariaHidden":65},[64],[35,5320,5322,5325,5328,5331,5334],{"className":5321},[69],[35,5323],{"className":5324,"style":180},[73],[35,5326,334],{"className":5327},[78,79],[35,5329],{"className":5330,"style":166},[165],[35,5332,136],{"className":5333},[170],[35,5335],{"className":5336,"style":166},[165],[35,5338,5340,5343,5383,5488,5491,5494],{"className":5339},[69],[35,5341],{"className":5342,"style":5004},[73],[35,5344,5346,5349],{"className":5345},[78],[35,5347,139],{"className":5348},[78,79],[35,5350,5352],{"className":5351},[363],[35,5353,5355,5375],{"className":5354},[367,2234],[35,5356,5358,5372],{"className":5357},[371],[35,5359,5361],{"className":5360,"style":2241},[375],[35,5362,5363,5366],{"style":2244},[35,5364],{"className":5365,"style":383},[382],[35,5367,5369],{"className":5368},[387,388,389,390],[35,5370,254],{"className":5371},[78,390],[35,5373,2258],{"className":5374},[2257],[35,5376,5378],{"className":5377},[371],[35,5379,5381],{"className":5380,"style":2265},[375],[35,5382],{},[35,5384,5386,5389,5485],{"className":5385},[78],[35,5387],{"className":5388},[278,4136],[35,5390,5392],{"className":5391},[4041],[35,5393,5395,5477],{"className":5394},[367,2234],[35,5396,5398,5474],{"className":5397},[371],[35,5399,5401,5415,5423],{"className":5400,"style":5023},[375],[35,5402,5403,5406],{"style":4152},[35,5404],{"className":5405,"style":4156},[382],[35,5407,5409],{"className":5408},[387,388,389,390],[35,5410,5412],{"className":5411},[78,390],[35,5413,4621],{"className":5414,"style":4636},[78,79,390],[35,5416,5417,5420],{"style":4211},[35,5418],{"className":5419,"style":4156},[382],[35,5421],{"className":5422,"style":4219},[4218],[35,5424,5425,5428],{"style":4222},[35,5426],{"className":5427,"style":4156},[382],[35,5429,5431],{"className":5430},[387,388,389,390],[35,5432,5434],{"className":5433},[78,390],[35,5435,5437,5440],{"className":5436},[78,390],[35,5438,199],{"className":5439,"style":215},[78,79,390],[35,5441,5443],{"className":5442},[363],[35,5444,5446,5466],{"className":5445},[367,2234],[35,5447,5449,5463],{"className":5448},[371],[35,5450,5452],{"className":5451,"style":4181},[375],[35,5453,5454,5457],{"style":5077},[35,5455],{"className":5456,"style":4188},[382],[35,5458,5460],{"className":5459},[387,4192,4193,390],[35,5461,254],{"className":5462},[78,390],[35,5464,2258],{"className":5465},[2257],[35,5467,5469],{"className":5468},[371],[35,5470,5472],{"className":5471,"style":4206},[375],[35,5473],{},[35,5475,2258],{"className":5476},[2257],[35,5478,5480],{"className":5479},[371],[35,5481,5483],{"className":5482,"style":5107},[375],[35,5484],{},[35,5486],{"className":5487},[296,4136],[35,5489],{"className":5490,"style":881},[165],[35,5492,850],{"className":5493},[715],[35,5495],{"className":5496,"style":881},[165],[35,5498,5500,5503,5506,5509,5512],{"className":5499},[69],[35,5501],{"className":5502,"style":871},[73],[35,5504,5256],{"className":5505},[670],[35,5507],{"className":5508,"style":881},[165],[35,5510,850],{"className":5511},[715],[35,5513],{"className":5514,"style":881},[165],[35,5516,5518,5521,5561,5666,5669,5672],{"className":5517},[69],[35,5519],{"className":5520,"style":5004},[73],[35,5522,5524,5527],{"className":5523},[78],[35,5525,139],{"className":5526},[78,79],[35,5528,5530],{"className":5529},[363],[35,5531,5533,5553],{"className":5532},[367,2234],[35,5534,5536,5550],{"className":5535},[371],[35,5537,5539],{"className":5538,"style":4756},[375],[35,5540,5541,5544],{"style":2244},[35,5542],{"className":5543,"style":383},[382],[35,5545,5547],{"className":5546},[387,388,389,390],[35,5548,4087],{"className":5549},[78,79,390],[35,5551,2258],{"className":5552},[2257],[35,5554,5556],{"className":5555},[371],[35,5557,5559],{"className":5558,"style":2265},[375],[35,5560],{},[35,5562,5564,5567,5663],{"className":5563},[78],[35,5565],{"className":5566},[278,4136],[35,5568,5570],{"className":5569},[4041],[35,5571,5573,5655],{"className":5572},[367,2234],[35,5574,5576,5652],{"className":5575},[371],[35,5577,5579,5593,5601],{"className":5578,"style":5023},[375],[35,5580,5581,5584],{"style":4152},[35,5582],{"className":5583,"style":4156},[382],[35,5585,5587],{"className":5586},[387,388,389,390],[35,5588,5590],{"className":5589},[78,390],[35,5591,4621],{"className":5592,"style":4636},[78,79,390],[35,5594,5595,5598],{"style":4211},[35,5596],{"className":5597,"style":4156},[382],[35,5599],{"className":5600,"style":4219},[4218],[35,5602,5603,5606],{"style":4222},[35,5604],{"className":5605,"style":4156},[382],[35,5607,5609],{"className":5608},[387,388,389,390],[35,5610,5612],{"className":5611},[78,390],[35,5613,5615,5618],{"className":5614},[78,390],[35,5616,199],{"className":5617,"style":215},[78,79,390],[35,5619,5621],{"className":5620},[363],[35,5622,5624,5644],{"className":5623},[367,2234],[35,5625,5627,5641],{"className":5626},[371],[35,5628,5630],{"className":5629,"style":4502},[375],[35,5631,5632,5635],{"style":5077},[35,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",[35,5919,5921,5968],{"className":5920},[38],[35,5922,5924],{"className":5923},[42],[44,5925,5926],{"xmlns":46},[48,5927,5928,5965],{},[51,5929,5930,5933,5935,5941,5947,5949,5951,5953,5959],{},[54,5931,5932],{},"N",[134,5934,136],{},[2203,5936,5937,5939],{},[54,5938,139],{},[252,5940,254],{},[2203,5942,5943,5945],{},[54,5944,199],{},[252,5946,254],{},[134,5948,850],{},[134,5950,5256],{},[134,5952,850],{},[2203,5954,5955,5957],{},[54,5956,139],{},[54,5958,4087],{},[2203,5960,5961,5963],{},[54,5962,199],{},[54,5964,4087],{},[58,5966,5967],{"encoding":60},"N = n_1 k_1 + \\dots + n_m k_m",[35,5969,5971,5990,6085,6103],{"className":5970,"ariaHidden":65},[64],[35,5972,5974,5977,5981,5984,5987],{"className":5973},[69],[35,5975],{"className":5976,"style":74},[73],[35,5978,5932],{"className":5979,"style":5980},[78,79],"margin-right:0.109em;",[35,5982],{"className":5983,"style":166},[165],[35,5985,136],{"className":5986},[170],[35,5988],{"className":5989,"style":166},[165],[35,5991,5993,5996,6036,6076,6079,6082],{"className":5992},[69],[35,5994],{"className":5995,"style":5144},[73],[35,5997,5999,6002],{"className":5998},[78],[35,6000,139],{"className":6001},[78,79],[35,6003,6005],{"className":6004},[363],[35,6006,6008,6028],{"className":6007},[367,2234],[35,6009,6011,6025],{"className":6010},[371],[35,6012,6014],{"className":6013,"style":2241},[375],[35,6015,6016,6019],{"style":2244},[35,6017],{"className":6018,"style":383},[382],[35,6020,6022],{"className":6021},[387,388,389,390],[35,6023,254],{"className":6024},[78,390],[35,6026,2258],{"className":6027},[2257],[35,6029,6031],{"className":6030},[371],[35,6032,6034],{"className":6033,"style":2265},[375],[35,6035],{},[35,6037,6039,6042],{"className":6038},[78],[35,6040,199],{"className":6041,"style":215},[78,79],[35,6043,6045],{"className":6044},[363],[35,6046,6048,6068],{"className":6047},[367,2234],[35,6049,6051,6065],{"className":6050},[371],[35,6052,6054],{"className":6053,"style":2241},[375],[35,6055,6056,6059],{"style":5166},[35,6057],{"className":6058,"style":383},[382],[35,6060,6062],{"className":6061},[387,388,389,390],[35,6063,254],{"className":6064},[78,390],[35,6066,2258],{"className":6067},[2257],[35,6069,6071],{"className":6070},[371],[35,6072,6074],{"className":6073,"style":2265},[375],[35,6075],{},[35,6077],{"className":6078,"style":881},[165],[35,6080,850],{"className":6081},[715],[35,6083],{"className":6084,"style":881},[165],[35,6086,6088,6091,6094,6097,6100],{"className":6087},[69],[35,6089],{"className":6090,"style":871},[73],[35,6092,5256],{"className":6093},[670],[35,6095],{"className":6096,"style":881},[165],[35,6098,850],{"className":6099},[715],[35,6101],{"className":6102,"style":881},[165],[35,6104,6106,6109,6149],{"className":6105},[69],[35,6107],{"className":6108,"style":5144},[73],[35,6110,6112,6115],{"className":6111},[78],[35,6113,139],{"className":6114},[78,79],[35,6116,6118],{"className":6117},[363],[35,6119,6121,6141],{"className":6120},[367,2234],[35,6122,6124,6138],{"className":6123},[371],[35,6125,6127],{"className":6126,"style":4756},[375],[35,6128,6129,6132],{"style":2244},[35,6130],{"className":6131,"style":383},[382],[35,6133,6135],{"className":6134},[387,388,389,390],[35,6136,4087],{"className":6137},[78,79,390],[35,6139,2258],{"className":6140},[2257],[35,6142,6144],{"className":6143},[371],[35,6145,6147],{"className":6146,"style":2265},[375],[35,6148],{},[35,6150,6152,6155],{"className":6151},[78],[35,6153,199],{"className":6154,"style":215},[78,79],[35,6156,6158],{"className":6157},[363],[35,6159,6161,6181],{"className":6160},[367,2234],[35,6162,6164,6178],{"className":6163},[371],[35,6165,6167],{"className":6166,"style":4756},[375],[35,6168,6169,6172],{"style":5166},[35,6170],{"className":6171,"style":383},[382],[35,6173,6175],{"className":6174},[387,388,389,390],[35,6176,4087],{"className":6177},[78,79,390],[35,6179,2258],{"className":6180},[2257],[35,6182,6184],{"className":6183},[371],[35,6185,6187],{"className":6186,"style":2265},[375],[35,6188],{}," 肯定是个整数。",[464,6191,6192,6193,6221,6222,6333],{},"所以，整个子群 ",[35,6194,6196,6209],{"className":6195},[38],[35,6197,6199],{"className":6198},[42],[44,6200,6201],{"xmlns":46},[48,6202,6203,6207],{},[51,6204,6205],{},[54,6206,4005],{},[58,6208,4005],{"encoding":60},[35,6210,6212],{"className":6211,"ariaHidden":65},[64],[35,6213,6215,6218],{"className":6214},[69],[35,6216],{"className":6217,"style":74},[73],[35,6219,4005],{"className":6220,"style":4020},[78,79]," 实际上包含在群 ",[35,6223,6225,6247],{"className":6224},[38],[35,6226,6228],{"className":6227},[42],[44,6229,6230],{"xmlns":46},[48,6231,6232,6244],{},[51,6233,6234,6236,6242],{},[134,6235,427],{"stretchy":237},[4041,6237,6238,6240],{},[252,6239,254],{},[54,6241,4621],{},[134,6243,432],{"stretchy":237},[58,6245,6246],{"encoding":60},"\\langle \\frac{1}{S} \\rangle",[35,6248,6250],{"className":6249,"ariaHidden":65},[64],[35,6251,6253,6257,6260,6330],{"className":6252},[69],[35,6254],{"className":6255,"style":6256},[73],"height:1.1901em;vertical-align:-0.345em;",[35,6258,427],{"className":6259},[278],[35,6261,6263,6266,6327],{"className":6262},[78],[35,6264],{"className":6265},[278,4136],[35,6267,6269],{"className":6268},[4041],[35,6270,6272,6319],{"className":6271},[367,2234],[35,6273,6275,6316],{"className":6274},[371],[35,6276,6279,6293,6301],{"className":6277,"style":6278},[375],"height:0.8451em;",[35,6280,6281,6284],{"style":4152},[35,6282],{"className":6283,"style":4156},[382],[35,6285,6287],{"className":6286},[387,388,389,390],[35,6288,6290],{"className":6289},[78,390],[35,6291,4621],{"className":6292,"style":4636},[78,79,390],[35,6294,6295,6298],{"style":4211},[35,6296],{"className":6297,"style":4156},[382],[35,6299],{"className":6300,"style":4219},[4218],[35,6302,6304,6307],{"style":6303},"top:-3.394em;",[35,6305],{"className":6306,"style":4156},[382],[35,6308,6310],{"className":6309},[387,388,389,390],[35,6311,6313],{"className":6312},[78,390],[35,6314,254],{"className":6315},[78,390],[35,6317,2258],{"className":6318},[2257],[35,6320,6322],{"className":6321},[371],[35,6323,6325],{"className":6324,"style":5107},[375],[35,6326],{},[35,6328],{"className":6329},[296,4136],[35,6331,432],{"className":6332},[296]," 里面。",[464,6335,6336,6339],{},[30,6337,6338],{},"关键定理","：循环群的子群必定是循环群。",[464,6341,6342,6449,6450,6481,6482,6510,6511,6539],{},[35,6343,6345,6366],{"className":6344},[38],[35,6346,6348],{"className":6347},[42],[44,6349,6350],{"xmlns":46},[48,6351,6352,6364],{},[51,6353,6354,6356,6362],{},[134,6355,427],{"stretchy":237},[4041,6357,6358,6360],{},[252,6359,254],{},[54,6361,4621],{},[134,6363,432],{"stretchy":237},[58,6365,6246],{"encoding":60},[35,6367,6369],{"className":6368,"ariaHidden":65},[64],[35,6370,6372,6375,6378,6446],{"className":6371},[69],[35,6373],{"className":6374,"style":6256},[73],[35,6376,427],{"className":6377},[278],[35,6379,6381,6384,6443],{"className":6380},[78],[35,6382],{"className":6383},[278,4136],[35,6385,6387],{"className":6386},[4041],[35,6388,6390,6435],{"className":6389},[367,2234],[35,6391,6393,6432],{"className":6392},[371],[35,6394,6396,6410,6418],{"className":6395,"style":6278},[375],[35,6397,6398,6401],{"style":4152},[35,6399],{"className":6400,"style":4156},[382],[35,6402,6404],{"className":6403},[387,388,389,390],[35,6405,6407],{"className":6406},[78,390],[35,6408,4621],{"className":6409,"style":4636},[78,79,390],[35,6411,6412,6415],{"style":4211},[35,6413],{"className":6414,"style":4156},[382],[35,6416],{"className":6417,"style":4219},[4218],[35,6419,6420,6423],{"style":6303},[35,6421],{"className":6422,"style":4156},[382],[35,6424,6426],{"className":6425},[387,388,389,390],[35,6427,6429],{"className":6428},[78,390],[35,6430,254],{"className":6431},[78,390],[35,6433,2258],{"className":6434},[2257],[35,6436,6438],{"className":6437},[371],[35,6439,6441],{"className":6440,"style":5107},[375],[35,6442],{},[35,6444],{"className":6445},[296,4136],[35,6447,432],{"className":6448},[296]," 同构于整数加法群 ",[35,6451,6453,6468],{"className":6452},[38],[35,6454,6456],{"className":6455},[42],[44,6457,6458],{"xmlns":46},[48,6459,6460,6465],{},[51,6461,6462],{},[54,6463,6464],{"mathvariant":3928},"Z",[58,6466,6467],{"encoding":60},"\\mathbb{Z}",[35,6469,6471],{"className":6470,"ariaHidden":65},[64],[35,6472,6474,6478],{"className":6473},[69],[35,6475],{"className":6476,"style":6477},[73],"height:0.6889em;",[35,6479,6464],{"className":6480},[78,3954],"（它是循环群）。既然 ",[35,6483,6485,6498],{"className":6484},[38],[35,6486,6488],{"className":6487},[42],[44,6489,6490],{"xmlns":46},[48,6491,6492,6496],{},[51,6493,6494],{},[54,6495,4005],{},[58,6497,4005],{"encoding":60},[35,6499,6501],{"className":6500,"ariaHidden":65},[64],[35,6502,6504,6507],{"className":6503},[69],[35,6505],{"className":6506,"style":74},[73],[35,6508,4005],{"className":6509,"style":4020},[78,79]," 是它的子群，那 ",[35,6512,6514,6527],{"className":6513},[38],[35,6515,6517],{"className":6516},[42],[44,6518,6519],{"xmlns":46},[48,6520,6521,6525],{},[51,6522,6523],{},[54,6524,4005],{},[58,6526,4005],{"encoding":60},[35,6528,6530],{"className":6529,"ariaHidden":65},[64],[35,6531,6533,6536],{"className":6532},[69],[35,6534],{"className":6535,"style":74},[73],[35,6537,4005],{"className":6538,"style":4020},[78,79]," 也一定是循环群。",[14,6541],{},[17,6543,6545,6546,6618,6619,6690],{"id":6544},"第二部分置换群-sns_nsn-与-ana_nan计算篇","第二部分：置换群 ",[35,6547,6549,6567],{"className":6548},[38],[35,6550,6552],{"className":6551},[42],[44,6553,6554],{"xmlns":46},[48,6555,6556,6564],{},[51,6557,6558],{},[2203,6559,6560,6562],{},[54,6561,4621],{},[54,6563,139],{},[58,6565,6566],{"encoding":60},"S_n",[35,6568,6570],{"className":6569,"ariaHidden":65},[64],[35,6571,6573,6577],{"className":6572},[69],[35,6574],{"className":6575,"style":6576},[73],"height:0.8333em;vertical-align:-0.15em;",[35,6578,6580,6583],{"className":6579},[78],[35,6581,4621],{"className":6582,"style":4636},[78,79],[35,6584,6586],{"className":6585},[363],[35,6587,6589,6610],{"className":6588},[367,2234],[35,6590,6592,6607],{"className":6591},[371],[35,6593,6595],{"className":6594,"style":4756},[375],[35,6596,6598,6601],{"style":6597},"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;",[35,6599],{"className":6600,"style":383},[382],[35,6602,6604],{"className":6603},[387,388,389,390],[35,6605,139],{"className":6606},[78,79,390],[35,6608,2258],{"className":6609},[2257],[35,6611,6613],{"className":6612},[371],[35,6614,6616],{"className":6615,"style":2265},[375],[35,6617],{}," 与 ",[35,6620,6622,6641],{"className":6621},[38],[35,6623,6625],{"className":6624},[42],[44,6626,6627],{"xmlns":46},[48,6628,6629,6638],{},[51,6630,6631],{},[2203,6632,6633,6636],{},[54,6634,6635],{},"A",[54,6637,139],{},[58,6639,6640],{"encoding":60},"A_n",[35,6642,6644],{"className":6643,"ariaHidden":65},[64],[35,6645,6647,6650],{"className":6646},[69],[35,6648],{"className":6649,"style":6576},[73],[35,6651,6653,6656],{"className":6652},[78],[35,6654,6635],{"className":6655},[78,79],[35,6657,6659],{"className":6658},[363],[35,6660,6662,6682],{"className":6661},[367,2234],[35,6663,6665,6679],{"className":6664},[371],[35,6666,6668],{"className":6667,"style":4756},[375],[35,6669,6670,6673],{"style":2244},[35,6671],{"className":6672,"style":383},[382],[35,6674,6676],{"className":6675},[387,388,389,390],[35,6677,139],{"className":6678},[78,79,390],[35,6680,2258],{"className":6681},[2257],[35,6683,6685],{"className":6684},[371],[35,6686,6688],{"className":6687,"style":2265},[375],[35,6689],{},"（计算篇）",[27,6692,6693,6696,6697,6820,6821,6849,6850,722],{},[30,6694,6695],{},"核心技巧","：我们要学会用“共轭”来搬运数字。\n公式：",[35,6698,6700,6744],{"className":6699},[38],[35,6701,6703],{"className":6702},[42],[44,6704,6705],{"xmlns":46},[48,6706,6707,6741],{},[51,6708,6709,6712,6714,6716,6719,6722,6724,6726,6729,6731],{},[54,6710,6711],{},"τ",[134,6713,238],{"stretchy":237},[54,6715,4806],{},[3629,6717,6718],{}," ",[54,6720,6721],{},"j",[3629,6723,6718],{},[134,6725,578],{},[3629,6727,6728],{}," ",[134,6730,248],{"stretchy":237},[330,6732,6733,6735],{},[54,6734,6711],{},[51,6736,6737,6739],{},[134,6738,591],{},[252,6740,254],{},[58,6742,6743],{"encoding":60},"\\tau (i \\ j \\ \\dots) \\tau^{-1}",[35,6745,6747],{"className":6746,"ariaHidden":65},[64],[35,6748,6750,6753,6757,6760,6763,6766,6770,6773,6776,6779,6782,6785],{"className":6749},[69],[35,6751],{"className":6752,"style":609},[73],[35,6754,6711],{"className":6755,"style":6756},[78,79],"margin-right:0.1132em;",[35,6758,238],{"className":6759},[278],[35,6761,4806],{"className":6762},[78,79],[35,6764,6718],{"className":6765},[165],[35,6767,6721],{"className":6768,"style":6769},[78,79],"margin-right:0.0572em;",[35,6771,6718],{"className":6772},[165],[35,6774],{"className":6775,"style":289},[165],[35,6777,578],{"className":6778},[670],[35,6780],{"className":6781,"style":289},[165],[35,6783,248],{"className":6784},[296],[35,6786,6788,6791],{"className":6787},[78],[35,6789,6711],{"className":6790,"style":6756},[78,79],[35,6792,6794],{"className":6793},[363],[35,6795,6797],{"className":6796},[367],[35,6798,6800],{"className":6799},[371],[35,6801,6803],{"className":6802,"style":649},[375],[35,6804,6805,6808],{"style":378},[35,6806],{"className":6807,"style":383},[382],[35,6809,6811],{"className":6810},[387,388,389,390],[35,6812,6814,6817],{"className":6813},[78,390],[35,6815,591],{"className":6816},[78,390],[35,6818,254],{"className":6819},[78,390]," 的结果，就是把括号里的数字 ",[35,6822,6824,6837],{"className":6823},[38],[35,6825,6827],{"className":6826},[42],[44,6828,6829],{"xmlns":46},[48,6830,6831,6835],{},[51,6832,6833],{},[54,6834,334],{},[58,6836,334],{"encoding":60},[35,6838,6840],{"className":6839,"ariaHidden":65},[64],[35,6841,6843,6846],{"className":6842},[69],[35,6844],{"className":6845,"style":180},[73],[35,6847,334],{"className":6848},[78,79]," 变成 ",[35,6851,6853,6873],{"className":6852},[38],[35,6854,6856],{"className":6855},[42],[44,6857,6858],{"xmlns":46},[48,6859,6860,6870],{},[51,6861,6862,6864,6866,6868],{},[54,6863,6711],{},[134,6865,238],{"stretchy":237},[54,6867,334],{},[134,6869,248],{"stretchy":237},[58,6871,6872],{"encoding":60},"\\tau(x)",[35,6874,6876],{"className":6875,"ariaHidden":65},[64],[35,6877,6879,6882,6885,6888,6891],{"className":6878},[69],[35,6880],{"className":6881,"style":152},[73],[35,6883,6711],{"className":6884,"style":6756},[78,79],[35,6886,238],{"className":6887},[278],[35,6889,334],{"className":6890},[78,79],[35,6892,248],{"className":6893},[296],[22,6895,6897,6898,6967],{"id":6896},"题目-11sns_nsn-的生成元","题目 11：",[35,6899,6901,6918],{"className":6900},[38],[35,6902,6904],{"className":6903},[42],[44,6905,6906],{"xmlns":46},[48,6907,6908,6916],{},[51,6909,6910],{},[2203,6911,6912,6914],{},[54,6913,4621],{},[54,6915,139],{},[58,6917,6566],{"encoding":60},[35,6919,6921],{"className":6920,"ariaHidden":65},[64],[35,6922,6924,6927],{"className":6923},[69],[35,6925],{"className":6926,"style":6576},[73],[35,6928,6930,6933],{"className":6929},[78],[35,6931,4621],{"className":6932,"style":4636},[78,79],[35,6934,6936],{"className":6935},[363],[35,6937,6939,6959],{"className":6938},[367,2234],[35,6940,6942,6956],{"className":6941},[371],[35,6943,6945],{"className":6944,"style":4756},[375],[35,6946,6947,6950],{"style":6597},[35,6948],{"className":6949,"style":383},[382],[35,6951,6953],{"className":6952},[387,388,389,390],[35,6954,139],{"className":6955},[78,79,390],[35,6957,2258],{"className":6958},[2257],[35,6960,6962],{"className":6961},[371],[35,6963,6965],{"className":6964,"style":2265},[375],[35,6966],{}," 的生成元",[27,6969,6970,6972,6973,7042,7043,7092,7093,7191],{},[30,6971,32],{}," 证明 ",[35,6974,6976,6993],{"className":6975},[38],[35,6977,6979],{"className":6978},[42],[44,6980,6981],{"xmlns":46},[48,6982,6983,6991],{},[51,6984,6985],{},[2203,6986,6987,6989],{},[54,6988,4621],{},[54,6990,139],{},[58,6992,6566],{"encoding":60},[35,6994,6996],{"className":6995,"ariaHidden":65},[64],[35,6997,6999,7002],{"className":6998},[69],[35,7000],{"className":7001,"style":6576},[73],[35,7003,7005,7008],{"className":7004},[78],[35,7006,4621],{"className":7007,"style":4636},[78,79],[35,7009,7011],{"className":7010},[363],[35,7012,7014,7034],{"className":7013},[367,2234],[35,7015,7017,7031],{"className":7016},[371],[35,7018,7020],{"className":7019,"style":4756},[375],[35,7021,7022,7025],{"style":6597},[35,7023],{"className":7024,"style":383},[382],[35,7026,7028],{"className":7027},[387,388,389,390],[35,7029,139],{"className":7030},[78,79,390],[35,7032,2258],{"className":7033},[2257],[35,7035,7037],{"className":7036},[371],[35,7038,7040],{"className":7039,"style":2265},[375],[35,7041],{}," 可以由对换 ",[35,7044,7046,7068],{"className":7045},[38],[35,7047,7049],{"className":7048},[42],[44,7050,7051],{"xmlns":46},[48,7052,7053,7065],{},[51,7054,7055,7057,7059,7061,7063],{},[134,7056,238],{"stretchy":237},[252,7058,254],{},[3629,7060,6718],{},[252,7062,573],{},[134,7064,248],{"stretchy":237},[58,7066,7067],{"encoding":60},"(1\\ 2)",[35,7069,7071],{"className":7070,"ariaHidden":65},[64],[35,7072,7074,7077,7080,7083,7086,7089],{"className":7073},[69],[35,7075],{"className":7076,"style":152},[73],[35,7078,238],{"className":7079},[278],[35,7081,254],{"className":7082},[78],[35,7084,6718],{"className":7085},[165],[35,7087,573],{"className":7088},[78],[35,7090,248],{"className":7091},[296]," 和轮换 ",[35,7094,7096,7131],{"className":7095},[38],[35,7097,7099],{"className":7098},[42],[44,7100,7101],{"xmlns":46},[48,7102,7103,7128],{},[51,7104,7105,7108,7110,7112,7114,7116,7118,7120,7122,7124,7126],{},[54,7106,7107],{},"σ",[134,7109,136],{},[134,7111,238],{"stretchy":237},[252,7113,254],{},[3629,7115,6718],{},[252,7117,573],{},[3629,7119,6718],{},[134,7121,5256],{},[3629,7123,6718],{},[54,7125,139],{},[134,7127,248],{"stretchy":237},[58,7129,7130],{"encoding":60},"\\sigma = (1\\ 2\\ \\cdots\\ n)",[35,7132,7134,7152],{"className":7133,"ariaHidden":65},[64],[35,7135,7137,7140,7143,7146,7149],{"className":7136},[69],[35,7138],{"className":7139,"style":180},[73],[35,7141,7107],{"className":7142,"style":111},[78,79],[35,7144],{"className":7145,"style":166},[165],[35,7147,136],{"className":7148},[170],[35,7150],{"className":7151,"style":166},[165],[35,7153,7155,7158,7161,7164,7167,7170,7173,7176,7179,7182,7185,7188],{"className":7154},[69],[35,7156],{"className":7157,"style":152},[73],[35,7159,238],{"className":7160},[278],[35,7162,254],{"className":7163},[78],[35,7165,6718],{"className":7166},[165],[35,7168,573],{"className":7169},[78],[35,7171,6718],{"className":7172},[165],[35,7174],{"className":7175,"style":289},[165],[35,7177,5256],{"className":7178},[670],[35,7180,6718],{"className":7181},[165],[35,7183],{"className":7184,"style":289},[165],[35,7186,139],{"className":7187},[78,79],[35,7189,248],{"className":7190},[296]," 生成。",[27,7193,7194,7197,7198,7272,7273,722],{},[30,7195,7196],{},"【证明思路】","\n只要能造出所有的“相邻对换” ",[35,7199,7201,7227],{"className":7200},[38],[35,7202,7204],{"className":7203},[42],[44,7205,7206],{"xmlns":46},[48,7207,7208,7224],{},[51,7209,7210,7212,7214,7216,7218,7220,7222],{},[134,7211,238],{"stretchy":237},[54,7213,199],{},[134,7215,243],{"separator":65},[54,7217,199],{},[134,7219,850],{},[252,7221,254],{},[134,7223,248],{"stretchy":237},[58,7225,7226],{"encoding":60},"(k, k+1)",[35,7228,7230,7260],{"className":7229,"ariaHidden":65},[64],[35,7231,7233,7236,7239,7242,7245,7248,7251,7254,7257],{"className":7232},[69],[35,7234],{"className":7235,"style":152},[73],[35,7237,238],{"className":7238},[278],[35,7240,199],{"className":7241,"style":215},[78,79],[35,7243,243],{"className":7244},[285],[35,7246],{"className":7247,"style":289},[165],[35,7249,199],{"className":7250,"style":215},[78,79],[35,7252],{"className":7253,"style":881},[165],[35,7255,850],{"className":7256},[715],[35,7258],{"className":7259,"style":881},[165],[35,7261,7263,7266,7269],{"className":7262},[69],[35,7264],{"className":7265,"style":152},[73],[35,7267,254],{"className":7268},[78],[35,7270,248],{"className":7271},[296],"，就能生成整个 ",[35,7274,7276,7293],{"className":7275},[38],[35,7277,7279],{"className":7278},[42],[44,7280,7281],{"xmlns":46},[48,7282,7283,7291],{},[51,7284,7285],{},[2203,7286,7287,7289],{},[54,7288,4621],{},[54,7290,139],{},[58,7292,6566],{"encoding":60},[35,7294,7296],{"className":7295,"ariaHidden":65},[64],[35,7297,7299,7302],{"className":7298},[69],[35,7300],{"className":7301,"style":6576},[73],[35,7303,7305,7308],{"className":7304},[78],[35,7306,4621],{"className":7307,"style":4636},[78,79],[35,7309,7311],{"className":7310},[363],[35,7312,7314,7334],{"className":7313},[367,2234],[35,7315,7317,7331],{"className":7316},[371],[35,7318,7320],{"className":7319,"style":4756},[375],[35,7321,7322,7325],{"style":6597},[35,7323],{"className":7324,"style":383},[382],[35,7326,7328],{"className":7327},[387,388,389,390],[35,7329,139],{"className":7330},[78,79,390],[35,7332,2258],{"className":7333},[2257],[35,7335,7337],{"className":7336},[371],[35,7338,7340],{"className":7339,"style":2265},[375],[35,7341],{},[461,7343,7344,7395,7775,7920,8076],{},[464,7345,7346,7347,722],{},"我们已有 ",[35,7348,7350,7371],{"className":7349},[38],[35,7351,7353],{"className":7352},[42],[44,7354,7355],{"xmlns":46},[48,7356,7357,7369],{},[51,7358,7359,7361,7363,7365,7367],{},[134,7360,238],{"stretchy":237},[252,7362,254],{},[3629,7364,6718],{},[252,7366,573],{},[134,7368,248],{"stretchy":237},[58,7370,7067],{"encoding":60},[35,7372,7374],{"className":7373,"ariaHidden":65},[64],[35,7375,7377,7380,7383,7386,7389,7392],{"className":7376},[69],[35,7378],{"className":7379,"style":152},[73],[35,7381,238],{"className":7382},[278],[35,7384,254],{"className":7385},[78],[35,7387,6718],{"className":7388},[165],[35,7390,573],{"className":7391},[78],[35,7393,248],{"className":7394},[296],[464,7396,7397,7398,7497,7498,7527,7528,7580,7581,7633,7634,722],{},"利用共轭作用：",[35,7399,7401,7435],{"className":7400},[38],[35,7402,7404],{"className":7403},[42],[44,7405,7406],{"xmlns":46},[48,7407,7408,7432],{},[51,7409,7410,7412,7414,7416,7418,7420,7422],{},[54,7411,7107],{},[134,7413,238],{"stretchy":237},[252,7415,254],{},[3629,7417,6718],{},[252,7419,573],{},[134,7421,248],{"stretchy":237},[330,7423,7424,7426],{},[54,7425,7107],{},[51,7427,7428,7430],{},[134,7429,591],{},[252,7431,254],{},[58,7433,7434],{"encoding":60},"\\sigma (1\\ 2) \\sigma^{-1}",[35,7436,7438],{"className":7437,"ariaHidden":65},[64],[35,7439,7441,7444,7447,7450,7453,7456,7459,7462],{"className":7440},[69],[35,7442],{"className":7443,"style":609},[73],[35,7445,7107],{"className":7446,"style":111},[78,79],[35,7448,238],{"className":7449},[278],[35,7451,254],{"className":7452},[78],[35,7454,6718],{"className":7455},[165],[35,7457,573],{"className":7458},[78],[35,7460,248],{"className":7461},[296],[35,7463,7465,7468],{"className":7464},[78],[35,7466,7107],{"className":7467,"style":111},[78,79],[35,7469,7471],{"className":7470},[363],[35,7472,7474],{"className":7473},[367],[35,7475,7477],{"className":7476},[371],[35,7478,7480],{"className":7479,"style":649},[375],[35,7481,7482,7485],{"style":378},[35,7483],{"className":7484,"style":383},[382],[35,7486,7488],{"className":7487},[387,388,389,390],[35,7489,7491,7494],{"className":7490},[78,390],[35,7492,591],{"className":7493},[78,390],[35,7495,254],{"className":7496},[78,390],"。\n",[35,7499,7501,7515],{"className":7500},[38],[35,7502,7504],{"className":7503},[42],[44,7505,7506],{"xmlns":46},[48,7507,7508,7512],{},[51,7509,7510],{},[54,7511,7107],{},[58,7513,7514],{"encoding":60},"\\sigma",[35,7516,7518],{"className":7517,"ariaHidden":65},[64],[35,7519,7521,7524],{"className":7520},[69],[35,7522],{"className":7523,"style":180},[73],[35,7525,7107],{"className":7526,"style":111},[78,79]," 把 ",[35,7529,7531,7550],{"className":7530},[38],[35,7532,7534],{"className":7533},[42],[44,7535,7536],{"xmlns":46},[48,7537,7538,7547],{},[51,7539,7540,7542,7545],{},[252,7541,254],{},[134,7543,7544],{},"→",[252,7546,573],{},[58,7548,7549],{"encoding":60},"1 \\to 2",[35,7551,7553,7571],{"className":7552,"ariaHidden":65},[64],[35,7554,7556,7559,7562,7565,7568],{"className":7555},[69],[35,7557],{"className":7558,"style":312},[73],[35,7560,254],{"className":7561},[78],[35,7563],{"className":7564,"style":166},[165],[35,7566,7544],{"className":7567},[170],[35,7569],{"className":7570,"style":166},[165],[35,7572,7574,7577],{"className":7573},[69],[35,7575],{"className":7576,"style":312},[73],[35,7578,573],{"className":7579},[78],"，把 ",[35,7582,7584,7603],{"className":7583},[38],[35,7585,7587],{"className":7586},[42],[44,7588,7589],{"xmlns":46},[48,7590,7591,7600],{},[51,7592,7593,7595,7597],{},[252,7594,573],{},[134,7596,7544],{},[252,7598,7599],{},"3",[58,7601,7602],{"encoding":60},"2 \\to 3",[35,7604,7606,7624],{"className":7605,"ariaHidden":65},[64],[35,7607,7609,7612,7615,7618,7621],{"className":7608},[69],[35,7610],{"className":7611,"style":312},[73],[35,7613,573],{"className":7614},[78],[35,7616],{"className":7617,"style":166},[165],[35,7619,7544],{"className":7620},[170],[35,7622],{"className":7623,"style":166},[165],[35,7625,7627,7630],{"className":7626},[69],[35,7628],{"className":7629,"style":312},[73],[35,7631,7599],{"className":7632},[78],"。所以 ",[35,7635,7637,7683],{"className":7636},[38],[35,7638,7640],{"className":7639},[42],[44,7641,7642],{"xmlns":46},[48,7643,7644,7680],{},[51,7645,7646,7648,7650,7652,7654,7656,7658,7668,7670,7672,7674,7676,7678],{},[54,7647,7107],{},[134,7649,238],{"stretchy":237},[252,7651,254],{},[3629,7653,6718],{},[252,7655,573],{},[134,7657,248],{"stretchy":237},[330,7659,7660,7662],{},[54,7661,7107],{},[51,7663,7664,7666],{},[134,7665,591],{},[252,7667,254],{},[134,7669,136],{},[134,7671,238],{"stretchy":237},[252,7673,573],{},[3629,7675,6718],{},[252,7677,7599],{},[134,7679,248],{"stretchy":237},[58,7681,7682],{"encoding":60},"\\sigma (1\\ 2) \\sigma^{-1} = (2\\ 3)",[35,7684,7686,7754],{"className":7685,"ariaHidden":65},[64],[35,7687,7689,7692,7695,7698,7701,7704,7707,7710,7745,7748,7751],{"className":7688},[69],[35,7690],{"className":7691,"style":609},[73],[35,7693,7107],{"className":7694,"style":111},[78,79],[35,7696,238],{"className":7697},[278],[35,7699,254],{"className":7700},[78],[35,7702,6718],{"className":7703},[165],[35,7705,573],{"className":7706},[78],[35,7708,248],{"className":7709},[296],[35,7711,7713,7716],{"className":7712},[78],[35,7714,7107],{"className":7715,"style":111},[78,79],[35,7717,7719],{"className":7718},[363],[35,7720,7722],{"className":7721},[367],[35,7723,7725],{"className":7724},[371],[35,7726,7728],{"className":7727,"style":649},[375],[35,7729,7730,7733],{"style":378},[35,7731],{"className":7732,"style":383},[382],[35,7734,7736],{"className":7735},[387,388,389,390],[35,7737,7739,7742],{"className":7738},[78,390],[35,7740,591],{"className":7741},[78,390],[35,7743,254],{"className":7744},[78,390],[35,7746],{"className":7747,"style":166},[165],[35,7749,136],{"className":7750},[170],[35,7752],{"className":7753,"style":166},[165],[35,7755,7757,7760,7763,7766,7769,7772],{"className":7756},[69],[35,7758],{"className":7759,"style":152},[73],[35,7761,238],{"className":7762},[278],[35,7764,573],{"className":7765},[78],[35,7767,6718],{"className":7768},[165],[35,7770,7599],{"className":7771},[78],[35,7773,248],{"className":7774},[296],[464,7776,7777,7778,722],{},"再搞一次：",[35,7779,7781,7828],{"className":7780},[38],[35,7782,7784],{"className":7783},[42],[44,7785,7786],{"xmlns":46},[48,7787,7788,7825],{},[51,7789,7790,7792,7794,7796,7798,7800,7802,7812,7814,7816,7818,7820,7823],{},[54,7791,7107],{},[134,7793,238],{"stretchy":237},[252,7795,573],{},[3629,7797,6718],{},[252,7799,7599],{},[134,7801,248],{"stretchy":237},[330,7803,7804,7806],{},[54,7805,7107],{},[51,7807,7808,7810],{},[134,7809,591],{},[252,7811,254],{},[134,7813,136],{},[134,7815,238],{"stretchy":237},[252,7817,7599],{},[3629,7819,6718],{},[252,7821,7822],{},"4",[134,7824,248],{"stretchy":237},[58,7826,7827],{"encoding":60},"\\sigma (2\\ 3) \\sigma^{-1} = (3\\ 4)",[35,7829,7831,7899],{"className":7830,"ariaHidden":65},[64],[35,7832,7834,7837,7840,7843,7846,7849,7852,7855,7890,7893,7896],{"className":7833},[69],[35,7835],{"className":7836,"style":609},[73],[35,7838,7107],{"className":7839,"style":111},[78,79],[35,7841,238],{"className":7842},[278],[35,7844,573],{"className":7845},[78],[35,7847,6718],{"className":7848},[165],[35,7850,7599],{"className":7851},[78],[35,7853,248],{"className":7854},[296],[35,7856,7858,7861],{"className":7857},[78],[35,7859,7107],{"className":7860,"style":111},[78,79],[35,7862,7864],{"className":7863},[363],[35,7865,7867],{"className":7866},[367],[35,7868,7870],{"className":7869},[371],[35,7871,7873],{"className":7872,"style":649},[375],[35,7874,7875,7878],{"style":378},[35,7876],{"className":7877,"style":383},[382],[35,7879,7881],{"className":7880},[387,388,389,390],[35,7882,7884,7887],{"className":7883},[78,390],[35,7885,591],{"className":7886},[78,390],[35,7888,254],{"className":7889},[78,390],[35,7891],{"className":7892,"style":166},[165],[35,7894,136],{"className":7895},[170],[35,7897],{"className":7898,"style":166},[165],[35,7900,7902,7905,7908,7911,7914,7917],{"className":7901},[69],[35,7903],{"className":7904,"style":152},[73],[35,7906,238],{"className":7907},[278],[35,7909,7599],{"className":7910},[78],[35,7912,6718],{"className":7913},[165],[35,7915,7822],{"className":7916},[78],[35,7918,248],{"className":7919},[296],[464,7921,7922,7923,722],{},"以此类推，我们可以得到 ",[35,7924,7926,7980],{"className":7925},[38],[35,7927,7929],{"className":7928},[42],[44,7930,7931],{"xmlns":46},[48,7932,7933,7977],{},[51,7934,7935,7937,7939,7941,7943,7945,7947,7949,7951,7953,7955,7957,7959,7961,7963,7965,7967,7969,7971,7973,7975],{},[134,7936,238],{"stretchy":237},[252,7938,254],{},[3629,7940,6718],{},[252,7942,573],{},[134,7944,248],{"stretchy":237},[134,7946,243],{"separator":65},[134,7948,238],{"stretchy":237},[252,7950,573],{},[3629,7952,6718],{},[252,7954,7599],{},[134,7956,248],{"stretchy":237},[134,7958,243],{"separator":65},[134,7960,578],{},[134,7962,243],{"separator":65},[134,7964,238],{"stretchy":237},[54,7966,139],{},[134,7968,591],{},[252,7970,254],{},[3629,7972,6718],{},[54,7974,139],{},[134,7976,248],{"stretchy":237},[58,7978,7979],{"encoding":60},"(1\\ 2), (2\\ 3), \\dots, (n-1\\ n)",[35,7981,7983,8058],{"className":7982,"ariaHidden":65},[64],[35,7984,7986,7989,7992,7995,7998,8001,8004,8007,8010,8013,8016,8019,8022,8025,8028,8031,8034,8037,8040,8043,8046,8049,8052,8055],{"className":7985},[69],[35,7987],{"className":7988,"style":152},[73],[35,7990,238],{"className":7991},[278],[35,7993,254],{"className":7994},[78],[35,7996,6718],{"className":7997},[165],[35,7999,573],{"className":8000},[78],[35,8002,248],{"className":8003},[296],[35,8005,243],{"className":8006},[285],[35,8008],{"className":8009,"style":289},[165],[35,8011,238],{"className":8012},[278],[35,8014,573],{"className":8015},[78],[35,8017,6718],{"className":8018},[165],[35,8020,7599],{"className":8021},[78],[35,8023,248],{"className":8024},[296],[35,8026,243],{"className":8027},[285],[35,8029],{"className":8030,"style":289},[165],[35,8032,578],{"className":8033},[670],[35,8035],{"className":8036,"style":289},[165],[35,8038,243],{"className":8039},[285],[35,8041],{"className":8042,"style":289},[165],[35,8044,238],{"className":8045},[278],[35,8047,139],{"className":8048},[78,79],[35,8050],{"className":8051,"style":881},[165],[35,8053,591],{"className":8054},[715],[35,8056],{"className":8057,"style":881},[165],[35,8059,8061,8064,8067,8070,8073],{"className":8060},[69],[35,8062],{"className":8063,"style":152},[73],[35,8065,254],{"className":8066},[78],[35,8068,6718],{"className":8069},[165],[35,8071,139],{"className":8072},[78,79],[35,8074,248],{"className":8075},[296],[464,8077,8078],{},"既然有了所有相邻对换，它们就能生成所有置换。证毕。",[22,8080,8082,8083,6967],{"id":8081},"题目-12-13ana_nan-的生成元","题目 12 & 13：",[35,8084,8086,8103],{"className":8085},[38],[35,8087,8089],{"className":8088},[42],[44,8090,8091],{"xmlns":46},[48,8092,8093,8101],{},[51,8094,8095],{},[2203,8096,8097,8099],{},[54,8098,6635],{},[54,8100,139],{},[58,8102,6640],{"encoding":60},[35,8104,8106],{"className":8105,"ariaHidden":65},[64],[35,8107,8109,8112],{"className":8108},[69],[35,8110],{"className":8111,"style":6576},[73],[35,8113,8115,8118],{"className":8114},[78],[35,8116,6635],{"className":8117},[78,79],[35,8119,8121],{"className":8120},[363],[35,8122,8124,8144],{"className":8123},[367,2234],[35,8125,8127,8141],{"className":8126},[371],[35,8128,8130],{"className":8129,"style":4756},[375],[35,8131,8132,8135],{"style":2244},[35,8133],{"className":8134,"style":383},[382],[35,8136,8138],{"className":8137},[387,388,389,390],[35,8139,139],{"className":8140},[78,79,390],[35,8142,2258],{"className":8143},[2257],[35,8145,8147],{"className":8146},[371],[35,8148,8150],{"className":8149,"style":2265},[375],[35,8151],{},[27,8153,8154,8155,8224,8225,8284],{},"这道题利用了 ",[35,8156,8158,8175],{"className":8157},[38],[35,8159,8161],{"className":8160},[42],[44,8162,8163],{"xmlns":46},[48,8164,8165,8173],{},[51,8166,8167],{},[2203,8168,8169,8171],{},[54,8170,6635],{},[54,8172,139],{},[58,8174,6640],{"encoding":60},[35,8176,8178],{"className":8177,"ariaHidden":65},[64],[35,8179,8181,8184],{"className":8180},[69],[35,8182],{"className":8183,"style":6576},[73],[35,8185,8187,8190],{"className":8186},[78],[35,8188,6635],{"className":8189},[78,79],[35,8191,8193],{"className":8192},[363],[35,8194,8196,8216],{"className":8195},[367,2234],[35,8197,8199,8213],{"className":8198},[371],[35,8200,8202],{"className":8201,"style":4756},[375],[35,8203,8204,8207],{"style":2244},[35,8205],{"className":8206,"style":383},[382],[35,8208,8210],{"className":8209},[387,388,389,390],[35,8211,139],{"className":8212},[78,79,390],[35,8214,2258],{"className":8215},[2257],[35,8217,8219],{"className":8218},[371],[35,8220,8222],{"className":8221,"style":2265},[375],[35,8223],{},"（偶置换群）是由“3-轮换” ",[35,8226,8228,8254],{"className":8227},[38],[35,8229,8231],{"className":8230},[42],[44,8232,8233],{"xmlns":46},[48,8234,8235,8251],{},[51,8236,8237,8239,8241,8243,8245,8247,8249],{},[134,8238,238],{"stretchy":237},[54,8240,4806],{},[3629,8242,6718],{},[54,8244,6721],{},[3629,8246,6718],{},[54,8248,199],{},[134,8250,248],{"stretchy":237},[58,8252,8253],{"encoding":60},"(i\\ j\\ k)",[35,8255,8257],{"className":8256,"ariaHidden":65},[64],[35,8258,8260,8263,8266,8269,8272,8275,8278,8281],{"className":8259},[69],[35,8261],{"className":8262,"style":152},[73],[35,8264,238],{"className":8265},[278],[35,8267,4806],{"className":8268},[78,79],[35,8270,6718],{"className":8271},[165],[35,8273,6721],{"className":8274,"style":6769},[78,79],[35,8276,6718],{"className":8277},[165],[35,8279,199],{"className":8280,"style":215},[78,79],[35,8282,248],{"className":8283},[296]," 生成的事实。",[471,8286,8287,8349],{},[464,8288,8289,8348],{},[35,8290,8292,8318],{"className":8291},[38],[35,8293,8295],{"className":8294},[42],[44,8296,8297],{"xmlns":46},[48,8298,8299,8315],{},[51,8300,8301,8303,8305,8307,8309,8311,8313],{},[134,8302,238],{"stretchy":237},[252,8304,254],{},[3629,8306,6718],{},[252,8308,573],{},[3629,8310,6718],{},[252,8312,7599],{},[134,8314,248],{"stretchy":237},[58,8316,8317],{"encoding":60},"(1\\ 2\\ 3)",[35,8319,8321],{"className":8320,"ariaHidden":65},[64],[35,8322,8324,8327,8330,8333,8336,8339,8342,8345],{"className":8323},[69],[35,8325],{"className":8326,"style":152},[73],[35,8328,238],{"className":8329},[278],[35,8331,254],{"className":8332},[78],[35,8334,6718],{"className":8335},[165],[35,8337,573],{"className":8338},[78],[35,8340,6718],{"className":8341},[165],[35,8343,7599],{"className":8344},[78],[35,8346,248],{"className":8347},[296]," 是一个 3-轮换。",[464,8350,8351,8352,757,8410,8485,8486,8514],{},"如果你有 ",[35,8353,8355,8380],{"className":8354},[38],[35,8356,8358],{"className":8357},[42],[44,8359,8360],{"xmlns":46},[48,8361,8362,8378],{},[51,8363,8364,8366,8368,8370,8372,8374,8376],{},[134,8365,238],{"stretchy":237},[252,8367,254],{},[3629,8369,6718],{},[252,8371,573],{},[3629,8373,6718],{},[252,8375,7599],{},[134,8377,248],{"stretchy":237},[58,8379,8317],{"encoding":60},[35,8381,8383],{"className":8382,"ariaHidden":65},[64],[35,8384,8386,8389,8392,8395,8398,8401,8404,8407],{"className":8385},[69],[35,8387],{"className":8388,"style":152},[73],[35,8390,238],{"className":8391},[278],[35,8393,254],{"className":8394},[78],[35,8396,6718],{"className":8397},[165],[35,8399,573],{"className":8400},[78],[35,8402,6718],{"className":8403},[165],[35,8405,7599],{"className":8406},[78],[35,8408,248],{"className":8409},[296],[35,8411,8413,8443],{"className":8412},[38],[35,8414,8416],{"className":8415},[42],[44,8417,8418],{"xmlns":46},[48,8419,8420,8440],{},[51,8421,8422,8424,8426,8428,8430,8432,8434,8436,8438],{},[134,8423,238],{"stretchy":237},[252,8425,254],{},[3629,8427,6718],{},[252,8429,573],{},[3629,8431,6718],{},[134,8433,578],{},[3629,8435,6718],{},[54,8437,139],{},[134,8439,248],{"stretchy":237},[58,8441,8442],{"encoding":60},"(1\\ 2\\ \\dots\\ n)",[35,8444,8446],{"className":8445,"ariaHidden":65},[64],[35,8447,8449,8452,8455,8458,8461,8464,8467,8470,8473,8476,8479,8482],{"className":8448},[69],[35,8450],{"className":8451,"style":152},[73],[35,8453,238],{"className":8454},[278],[35,8456,254],{"className":8457},[78],[35,8459,6718],{"className":8460},[165],[35,8462,573],{"className":8463},[78],[35,8465,6718],{"className":8466},[165],[35,8468],{"className":8469,"style":289},[165],[35,8471,578],{"className":8472},[670],[35,8474,6718],{"className":8475},[165],[35,8477],{"className":8478,"style":289},[165],[35,8480,139],{"className":8481},[78,79],[35,8483,248],{"className":8484},[296],"（当 ",[35,8487,8489,8502],{"className":8488},[38],[35,8490,8492],{"className":8491},[42],[44,8493,8494],{"xmlns":46},[48,8495,8496,8500],{},[51,8497,8498],{},[54,8499,139],{},[58,8501,139],{"encoding":60},[35,8503,8505],{"className":8504,"ariaHidden":65},[64],[35,8506,8508,8511],{"className":8507},[69],[35,8509],{"className":8510,"style":180},[73],[35,8512,139],{"className":8513},[78,79]," 为奇数时，它是偶置换），你就可以像上面一样，通过“推移”造出所有的 3-轮换。",[14,8516],{},[17,8518,8520],{"id":8519},"第三部分正规子群与中心化子抽象篇","第三部分：正规子群与中心化子（抽象篇）",[22,8522,8524],{"id":8523},"题目-14中心化子与共轭类","题目 14：中心化子与共轭类",[27,8526,8527,8529,8530,8626,8627,8795,8796,8861],{},[30,8528,32],{}," 设 ",[35,8531,8533,8566],{"className":8532},[38],[35,8534,8536],{"className":8535},[42],[44,8537,8538],{"xmlns":46},[48,8539,8540,8564],{},[51,8541,8542,8544,8546,8548,8550,8552,8554,8556,8558,8560,8562],{},[54,8543,7107],{},[134,8545,136],{},[134,8547,238],{"stretchy":237},[252,8549,254],{},[3629,8551,6718],{},[252,8553,573],{},[3629,8555,6718],{},[134,8557,5256],{},[3629,8559,6718],{},[54,8561,139],{},[134,8563,248],{"stretchy":237},[58,8565,7130],{"encoding":60},[35,8567,8569,8587],{"className":8568,"ariaHidden":65},[64],[35,8570,8572,8575,8578,8581,8584],{"className":8571},[69],[35,8573],{"className":8574,"style":180},[73],[35,8576,7107],{"className":8577,"style":111},[78,79],[35,8579],{"className":8580,"style":166},[165],[35,8582,136],{"className":8583},[170],[35,8585],{"className":8586,"style":166},[165],[35,8588,8590,8593,8596,8599,8602,8605,8608,8611,8614,8617,8620,8623],{"className":8589},[69],[35,8591],{"className":8592,"style":152},[73],[35,8594,238],{"className":8595},[278],[35,8597,254],{"className":8598},[78],[35,8600,6718],{"className":8601},[165],[35,8603,573],{"className":8604},[78],[35,8606,6718],{"className":8607},[165],[35,8609],{"className":8610,"style":289},[165],[35,8612,5256],{"className":8613},[670],[35,8615,6718],{"className":8616},[165],[35,8618],{"className":8619,"style":289},[165],[35,8621,139],{"className":8622},[78,79],[35,8624,248],{"className":8625},[296],"，证明其中心化子 ",[35,8628,8630,8667],{"className":8629},[38],[35,8631,8633],{"className":8632},[42],[44,8634,8635],{"xmlns":46},[48,8636,8637,8664],{},[51,8638,8639,8650,8652,8654,8656,8658,8660,8662],{},[2203,8640,8641,8644],{},[54,8642,8643],{},"C",[2203,8645,8646,8648],{},[54,8647,4621],{},[54,8649,139],{},[134,8651,238],{"stretchy":237},[54,8653,7107],{},[134,8655,248],{"stretchy":237},[134,8657,136],{},[134,8659,427],{"stretchy":237},[54,8661,7107],{},[134,8663,432],{"stretchy":237},[58,8665,8666],{"encoding":60},"C_{S_n}(\\sigma) = \\langle \\sigma \\rangle",[35,8668,8670,8780],{"className":8669,"ariaHidden":65},[64],[35,8671,8673,8677,8762,8765,8768,8771,8774,8777],{"className":8672},[69],[35,8674],{"className":8675,"style":8676},[73],"height:1.0001em;vertical-align:-0.2501em;",[35,8678,8680,8684],{"className":8679},[78],[35,8681,8643],{"className":8682,"style":8683},[78,79],"margin-right:0.0715em;",[35,8685,8687],{"className":8686},[363],[35,8688,8690,8753],{"className":8689},[367,2234],[35,8691,8693,8750],{"className":8692},[371],[35,8694,8697],{"className":8695,"style":8696},[375],"height:0.3283em;",[35,8698,8700,8703],{"style":8699},"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;",[35,8701],{"className":8702,"style":383},[382],[35,8704,8706],{"className":8705},[387,388,389,390],[35,8707,8709],{"className":8708},[78,390],[35,8710,8712,8715],{"className":8711},[78,390],[35,8713,4621],{"className":8714,"style":4636},[78,79,390],[35,8716,8718],{"className":8717},[363],[35,8719,8721,8742],{"className":8720},[367,2234],[35,8722,8724,8739],{"className":8723},[371],[35,8725,8727],{"className":8726,"style":4502},[375],[35,8728,8730,8733],{"style":8729},"top:-2.357em;margin-left:-0.0576em;margin-right:0.0714em;",[35,8731],{"className":8732,"style":4188},[382],[35,8734,8736],{"className":8735},[387,4192,4193,390],[35,8737,139],{"className":8738},[78,79,390],[35,8740,2258],{"className":8741},[2257],[35,8743,8745],{"className":8744},[371],[35,8746,8748],{"className":8747,"style":4206},[375],[35,8749],{},[35,8751,2258],{"className":8752},[2257],[35,8754,8756],{"className":8755},[371],[35,8757,8760],{"className":8758,"style":8759},[375],"height:0.2501em;",[35,8761],{},[35,8763,238],{"className":8764},[278],[35,8766,7107],{"className":8767,"style":111},[78,79],[35,8769,248],{"className":8770},[296],[35,8772],{"className":8773,"style":166},[165],[35,8775,136],{"className":8776},[170],[35,8778],{"className":8779,"style":166},[165],[35,8781,8783,8786,8789,8792],{"className":8782},[69],[35,8784],{"className":8785,"style":152},[73],[35,8787,427],{"className":8788},[278],[35,8790,7107],{"className":8791,"style":111},[78,79],[35,8793,432],{"className":8794},[296],"，共轭类含 ",[35,8797,8799,8824],{"className":8798},[38],[35,8800,8802],{"className":8801},[42],[44,8803,8804],{"xmlns":46},[48,8805,8806,8821],{},[51,8807,8808,8810,8812,8814,8816,8818],{},[134,8809,238],{"stretchy":237},[54,8811,139],{},[134,8813,591],{},[252,8815,254],{},[134,8817,248],{"stretchy":237},[134,8819,8820],{"stretchy":237},"!",[58,8822,8823],{"encoding":60},"(n-1)!",[35,8825,8827,8848],{"className":8826,"ariaHidden":65},[64],[35,8828,8830,8833,8836,8839,8842,8845],{"className":8829},[69],[35,8831],{"className":8832,"style":152},[73],[35,8834,238],{"className":8835},[278],[35,8837,139],{"className":8838},[78,79],[35,8840],{"className":8841,"style":881},[165],[35,8843,591],{"className":8844},[715],[35,8846],{"className":8847,"style":881},[165],[35,8849,8851,8854,8857],{"className":8850},[69],[35,8852],{"className":8853,"style":152},[73],[35,8855,254],{"className":8856},[78],[35,8858,8860],{"className":8859},[296],")!"," 个元素。",[27,8863,8864],{},[30,8865,8866],{},"【证明详解】",[461,8868,8869,9616,10466],{},[464,8870,8871,469,8874],{},[30,8872,8873],{},"共轭类的大小",[471,8875,8876,8990,9149,9421],{},[464,8877,8878,8879,8948,8949,8989],{},"在 ",[35,8880,8882,8899],{"className":8881},[38],[35,8883,8885],{"className":8884},[42],[44,8886,8887],{"xmlns":46},[48,8888,8889,8897],{},[51,8890,8891],{},[2203,8892,8893,8895],{},[54,8894,4621],{},[54,8896,139],{},[58,8898,6566],{"encoding":60},[35,8900,8902],{"className":8901,"ariaHidden":65},[64],[35,8903,8905,8908],{"className":8904},[69],[35,8906],{"className":8907,"style":6576},[73],[35,8909,8911,8914],{"className":8910},[78],[35,8912,4621],{"className":8913,"style":4636},[78,79],[35,8915,8917],{"className":8916},[363],[35,8918,8920,8940],{"className":8919},[367,2234],[35,8921,8923,8937],{"className":8922},[371],[35,8924,8926],{"className":8925,"style":4756},[375],[35,8927,8928,8931],{"style":6597},[35,8929],{"className":8930,"style":383},[382],[35,8932,8934],{"className":8933},[387,388,389,390],[35,8935,139],{"className":8936},[78,79,390],[35,8938,2258],{"className":8939},[2257],[35,8941,8943],{"className":8942},[371],[35,8944,8946],{"className":8945,"style":2265},[375],[35,8947],{}," 中，两个置换共轭 ",[35,8950,8952,8971],{"className":8951},[38],[35,8953,8955],{"className":8954},[42],[44,8956,8957],{"xmlns":46},[48,8958,8959,8968],{},[51,8960,8961,8963,8966],{},[3629,8962,3631],{},[134,8964,8965],{},"⟺",[3629,8967,3631],{},[58,8969,8970],{"encoding":60},"\\iff",[35,8972,8974],{"className":8973,"ariaHidden":65},[64],[35,8975,8977,8980,8983,8986],{"className":8976},[69],[35,8978],{"className":8979,"style":3775},[73],[35,8981],{"className":8982,"style":166},[165],[35,8984,8965],{"className":8985},[170],[35,8987],{"className":8988,"style":166},[165]," 它们的循环结构相同。",[464,8991,8992,9020,9021,9049,9050,9119,9120,9148],{},[35,8993,8995,9008],{"className":8994},[38],[35,8996,8998],{"className":8997},[42],[44,8999,9000],{"xmlns":46},[48,9001,9002,9006],{},[51,9003,9004],{},[54,9005,7107],{},[58,9007,7514],{"encoding":60},[35,9009,9011],{"className":9010,"ariaHidden":65},[64],[35,9012,9014,9017],{"className":9013},[69],[35,9015],{"className":9016,"style":180},[73],[35,9018,7107],{"className":9019,"style":111},[78,79]," 是一个 ",[35,9022,9024,9037],{"className":9023},[38],[35,9025,9027],{"className":9026},[42],[44,9028,9029],{"xmlns":46},[48,9030,9031,9035],{},[51,9032,9033],{},[54,9034,139],{},[58,9036,139],{"encoding":60},[35,9038,9040],{"className":9039,"ariaHidden":65},[64],[35,9041,9043,9046],{"className":9042},[69],[35,9044],{"className":9045,"style":180},[73],[35,9047,139],{"className":9048},[78,79],"-长轮换。在 ",[35,9051,9053,9070],{"className":9052},[38],[35,9054,9056],{"className":9055},[42],[44,9057,9058],{"xmlns":46},[48,9059,9060,9068],{},[51,9061,9062],{},[2203,9063,9064,9066],{},[54,9065,4621],{},[54,9067,139],{},[58,9069,6566],{"encoding":60},[35,9071,9073],{"className":9072,"ariaHidden":65},[64],[35,9074,9076,9079],{"className":9075},[69],[35,9077],{"className":9078,"style":6576},[73],[35,9080,9082,9085],{"className":9081},[78],[35,9083,4621],{"className":9084,"style":4636},[78,79],[35,9086,9088],{"className":9087},[363],[35,9089,9091,9111],{"className":9090},[367,2234],[35,9092,9094,9108],{"className":9093},[371],[35,9095,9097],{"className":9096,"style":4756},[375],[35,9098,9099,9102],{"style":6597},[35,9100],{"className":9101,"style":383},[382],[35,9103,9105],{"className":9104},[387,388,389,390],[35,9106,139],{"className":9107},[78,79,390],[35,9109,2258],{"className":9110},[2257],[35,9112,9114],{"className":9113},[371],[35,9115,9117],{"className":9116,"style":2265},[375],[35,9118],{}," 中，有多少个长为 ",[35,9121,9123,9136],{"className":9122},[38],[35,9124,9126],{"className":9125},[42],[44,9127,9128],{"xmlns":46},[48,9129,9130,9134],{},[51,9131,9132],{},[54,9133,139],{},[58,9135,139],{"encoding":60},[35,9137,9139],{"className":9138,"ariaHidden":65},[64],[35,9140,9142,9145],{"className":9141},[69],[35,9143],{"className":9144,"style":180},[73],[35,9146,139],{"className":9147},[78,79]," 的轮换？",[464,9150,9151,9152,9180,9181,9209,9210,9244,9245,757,9303,9362,9363,9391,9392,9420],{},"排列 ",[35,9153,9155,9168],{"className":9154},[38],[35,9156,9158],{"className":9157},[42],[44,9159,9160],{"xmlns":46},[48,9161,9162,9166],{},[51,9163,9164],{},[252,9165,254],{},[58,9167,254],{"encoding":60},[35,9169,9171],{"className":9170,"ariaHidden":65},[64],[35,9172,9174,9177],{"className":9173},[69],[35,9175],{"className":9176,"style":312},[73],[35,9178,254],{"className":9179},[78]," 到 ",[35,9182,9184,9197],{"className":9183},[38],[35,9185,9187],{"className":9186},[42],[44,9188,9189],{"xmlns":46},[48,9190,9191,9195],{},[51,9192,9193],{},[54,9194,139],{},[58,9196,139],{"encoding":60},[35,9198,9200],{"className":9199,"ariaHidden":65},[64],[35,9201,9203,9206],{"className":9202},[69],[35,9204],{"className":9205,"style":180},[73],[35,9207,139],{"className":9208},[78,79]," 有 ",[35,9211,9213,9229],{"className":9212},[38],[35,9214,9216],{"className":9215},[42],[44,9217,9218],{"xmlns":46},[48,9219,9220,9226],{},[51,9221,9222,9224],{},[54,9223,139],{},[134,9225,8820],{"stretchy":237},[58,9227,9228],{"encoding":60},"n!",[35,9230,9232],{"className":9231,"ariaHidden":65},[64],[35,9233,9235,9238,9241],{"className":9234},[69],[35,9236],{"className":9237,"style":211},[73],[35,9239,139],{"className":9240},[78,79],[35,9242,8820],{"className":9243},[296]," 种排法。但是轮换是圆的，",[35,9246,9248,9273],{"className":9247},[38],[35,9249,9251],{"className":9250},[42],[44,9252,9253],{"xmlns":46},[48,9254,9255,9271],{},[51,9256,9257,9259,9261,9263,9265,9267,9269],{},[134,9258,238],{"stretchy":237},[252,9260,254],{},[3629,9262,6718],{},[252,9264,573],{},[3629,9266,6718],{},[252,9268,7599],{},[134,9270,248],{"stretchy":237},[58,9272,8317],{"encoding":60},[35,9274,9276],{"className":9275,"ariaHidden":65},[64],[35,9277,9279,9282,9285,9288,9291,9294,9297,9300],{"className":9278},[69],[35,9280],{"className":9281,"style":152},[73],[35,9283,238],{"className":9284},[278],[35,9286,254],{"className":9287},[78],[35,9289,6718],{"className":9290},[165],[35,9292,573],{"className":9293},[78],[35,9295,6718],{"className":9296},[165],[35,9298,7599],{"className":9299},[78],[35,9301,248],{"className":9302},[296],[35,9304,9306,9332],{"className":9305},[38],[35,9307,9309],{"className":9308},[42],[44,9310,9311],{"xmlns":46},[48,9312,9313,9329],{},[51,9314,9315,9317,9319,9321,9323,9325,9327],{},[134,9316,238],{"stretchy":237},[252,9318,573],{},[3629,9320,6718],{},[252,9322,7599],{},[3629,9324,6718],{},[252,9326,254],{},[134,9328,248],{"stretchy":237},[58,9330,9331],{"encoding":60},"(2\\ 3\\ 1)",[35,9333,9335],{"className":9334,"ariaHidden":65},[64],[35,9336,9338,9341,9344,9347,9350,9353,9356,9359],{"className":9337},[69],[35,9339],{"className":9340,"style":152},[73],[35,9342,238],{"className":9343},[278],[35,9345,573],{"className":9346},[78],[35,9348,6718],{"className":9349},[165],[35,9351,7599],{"className":9352},[78],[35,9354,6718],{"className":9355},[165],[35,9357,254],{"className":9358},[78],[35,9360,248],{"className":9361},[296]," 是一样的。一个 ",[35,9364,9366,9379],{"className":9365},[38],[35,9367,9369],{"className":9368},[42],[44,9370,9371],{"xmlns":46},[48,9372,9373,9377],{},[51,9374,9375],{},[54,9376,139],{},[58,9378,139],{"encoding":60},[35,9380,9382],{"className":9381,"ariaHidden":65},[64],[35,9383,9385,9388],{"className":9384},[69],[35,9386],{"className":9387,"style":180},[73],[35,9389,139],{"className":9390},[78,79],"-轮换有 ",[35,9393,9395,9408],{"className":9394},[38],[35,9396,9398],{"className":9397},[42],[44,9399,9400],{"xmlns":46},[48,9401,9402,9406],{},[51,9403,9404],{},[54,9405,139],{},[58,9407,139],{"encoding":60},[35,9409,9411],{"className":9410,"ariaHidden":65},[64],[35,9412,9414,9417],{"className":9413},[69],[35,9415],{"className":9416,"style":180},[73],[35,9418,139],{"className":9419},[78,79]," 种写法。",[464,9422,9423,9424,9452,9453,722],{},"所以，",[35,9425,9427,9440],{"className":9426},[38],[35,9428,9430],{"className":9429},[42],[44,9431,9432],{"xmlns":46},[48,9433,9434,9438],{},[51,9435,9436],{},[54,9437,139],{},[58,9439,139],{"encoding":60},[35,9441,9443],{"className":9442,"ariaHidden":65},[64],[35,9444,9446,9449],{"className":9445},[69],[35,9447],{"className":9448,"style":180},[73],[35,9450,139],{"className":9451},[78,79],"-轮换的总数（共轭类大小） = ",[35,9454,9456,9492],{"className":9455},[38],[35,9457,9459],{"className":9458},[42],[44,9460,9461],{"xmlns":46},[48,9462,9463,9489],{},[51,9464,9465,9475,9477,9479,9481,9483,9485,9487],{},[4041,9466,9467,9473],{},[51,9468,9469,9471],{},[54,9470,139],{},[134,9472,8820],{"stretchy":237},[54,9474,139],{},[134,9476,136],{},[134,9478,238],{"stretchy":237},[54,9480,139],{},[134,9482,591],{},[252,9484,254],{},[134,9486,248],{"stretchy":237},[134,9488,8820],{"stretchy":237},[58,9490,9491],{"encoding":60},"\\frac{n!}{n} = (n-1)!",[35,9493,9495,9583,9604],{"className":9494,"ariaHidden":65},[64],[35,9496,9498,9502,9574,9577,9580],{"className":9497},[69],[35,9499],{"className":9500,"style":9501},[73],"height:1.2251em;vertical-align:-0.345em;",[35,9503,9505,9508,9571],{"className":9504},[78],[35,9506],{"className":9507},[278,4136],[35,9509,9511],{"className":9510},[4041],[35,9512,9514,9563],{"className":9513},[367,2234],[35,9515,9517,9560],{"className":9516},[371],[35,9518,9521,9535,9543],{"className":9519,"style":9520},[375],"height:0.8801em;",[35,9522,9523,9526],{"style":4152},[35,9524],{"className":9525,"style":4156},[382],[35,9527,9529],{"className":9528},[387,388,389,390],[35,9530,9532],{"className":9531},[78,390],[35,9533,139],{"className":9534},[78,79,390],[35,9536,9537,9540],{"style":4211},[35,9538],{"className":9539,"style":4156},[382],[35,9541],{"className":9542,"style":4219},[4218],[35,9544,9545,9548],{"style":6303},[35,9546],{"className":9547,"style":4156},[382],[35,9549,9551],{"className":9550},[387,388,389,390],[35,9552,9554,9557],{"className":9553},[78,390],[35,9555,139],{"className":9556},[78,79,390],[35,9558,8820],{"className":9559},[296,390],[35,9561,2258],{"className":9562},[2257],[35,9564,9566],{"className":9565},[371],[35,9567,9569],{"className":9568,"style":5107},[375],[35,9570],{},[35,9572],{"className":9573},[296,4136],[35,9575],{"className":9576,"style":166},[165],[35,9578,136],{"className":9579},[170],[35,9581],{"className":9582,"style":166},[165],[35,9584,9586,9589,9592,9595,9598,9601],{"className":9585},[69],[35,9587],{"className":9588,"style":152},[73],[35,9590,238],{"className":9591},[278],[35,9593,139],{"className":9594},[78,79],[35,9596],{"className":9597,"style":881},[165],[35,9599,591],{"className":9600},[715],[35,9602],{"className":9603,"style":881},[165],[35,9605,9607,9610,9613],{"className":9606},[69],[35,9608],{"className":9609,"style":152},[73],[35,9611,254],{"className":9612},[78],[35,9614,8860],{"className":9615},[296],[464,9617,9618,469,9621],{},[30,9619,9620],{},"轨道-稳定子定理（Orbit-Stabilizer Theorem）",[471,9622,9623,9786,9953,10178],{},[464,9624,9625,9626,722],{},"这是群论最重要的公式之一：",[35,9627,9629,9689],{"className":9628},[38],[35,9630,9632],{"className":9631},[42],[44,9633,9634],{"xmlns":46},[48,9635,9636,9686],{},[51,9637,9638,9640,9642,9644,9646,9648,9651,9653,9656,9658,9660,9662,9664,9666,9668,9670,9673,9676,9678,9680,9682,9684],{},[54,9639,128],{"mathvariant":127},[54,9641,56],{},[54,9643,128],{"mathvariant":127},[134,9645,136],{},[54,9647,128],{"mathvariant":127},[54,9649,9650],{},"O",[54,9652,4047],{},[54,9654,9655],{},"b",[134,9657,238],{"stretchy":237},[54,9659,7107],{},[134,9661,248],{"stretchy":237},[54,9663,128],{"mathvariant":127},[134,9665,1361],{},[54,9667,128],{"mathvariant":127},[54,9669,4621],{},[54,9671,9672],{},"t",[54,9674,9675],{},"a",[54,9677,9655],{},[134,9679,238],{"stretchy":237},[54,9681,7107],{},[134,9683,248],{"stretchy":237},[54,9685,128],{"mathvariant":127},[58,9687,9688],{"encoding":60},"|G| = |Orb(\\sigma)| \\cdot |Stab(\\sigma)|",[35,9690,9692,9716,9755],{"className":9691,"ariaHidden":65},[64],[35,9693,9695,9698,9701,9704,9707,9710,9713],{"className":9694},[69],[35,9696],{"className":9697,"style":152},[73],[35,9699,128],{"className":9700},[78],[35,9702,56],{"className":9703},[78,79],[35,9705,128],{"className":9706},[78],[35,9708],{"className":9709,"style":166},[165],[35,9711,136],{"className":9712},[170],[35,9714],{"className":9715,"style":166},[165],[35,9717,9719,9722,9725,9728,9731,9734,9737,9740,9743,9746,9749,9752],{"className":9718},[69],[35,9720],{"className":9721,"style":152},[73],[35,9723,128],{"className":9724},[78],[35,9726,9650],{"className":9727,"style":4238},[78,79],[35,9729,4047],{"className":9730,"style":4238},[78,79],[35,9732,9655],{"className":9733},[78,79],[35,9735,238],{"className":9736},[278],[35,9738,7107],{"className":9739,"style":111},[78,79],[35,9741,248],{"className":9742},[296],[35,9744,128],{"className":9745},[78],[35,9747],{"className":9748,"style":881},[165],[35,9750,1361],{"className":9751},[715],[35,9753],{"className":9754,"style":881},[165],[35,9756,9758,9761,9764,9767,9770,9774,9777,9780,9783],{"className":9757},[69],[35,9759],{"className":9760,"style":152},[73],[35,9762,128],{"className":9763},[78],[35,9765,4621],{"className":9766,"style":4636},[78,79],[35,9768,9672],{"className":9769},[78,79],[35,9771,9773],{"className":9772},[78,79],"ab",[35,9775,238],{"className":9776},[278],[35,9778,7107],{"className":9779,"style":111},[78,79],[35,9781,248],{"className":9782},[296],[35,9784,128],{"className":9785},[78],[464,9787,9788,9789,722],{},"在这里：",[35,9790,9792,9836],{"className":9791},[38],[35,9793,9795],{"className":9794},[42],[44,9796,9797],{"xmlns":46},[48,9798,9799,9833],{},[51,9800,9801,9803,9809,9811,9813,9815,9817,9820,9822,9824,9826,9828,9831],{},[54,9802,128],{"mathvariant":127},[2203,9804,9805,9807],{},[54,9806,4621],{},[54,9808,139],{},[54,9810,128],{"mathvariant":127},[134,9812,136],{},[134,9814,238],{"stretchy":237},[54,9816,7107],{},[3629,9818,9819],{},"的共轭类大小",[134,9821,248],{"stretchy":237},[134,9823,1361],{},[134,9825,238],{"stretchy":237},[54,9827,7107],{},[3629,9829,9830],{},"的中心化子阶数",[134,9832,248],{"stretchy":237},[58,9834,9835],{"encoding":60},"|S_n| = (\\sigma \\text{的共轭类大小}) \\cdot (\\sigma \\text{的中心化子阶数})",[35,9837,9839,9900,9932],{"className":9838,"ariaHidden":65},[64],[35,9840,9842,9845,9848,9888,9891,9894,9897],{"className":9841},[69],[35,9843],{"className":9844,"style":152},[73],[35,9846,128],{"className":9847},[78],[35,9849,9851,9854],{"className":9850},[78],[35,9852,4621],{"className":9853,"style":4636},[78,79],[35,9855,9857],{"className":9856},[363],[35,9858,9860,9880],{"className":9859},[367,2234],[35,9861,9863,9877],{"className":9862},[371],[35,9864,9866],{"className":9865,"style":4756},[375],[35,9867,9868,9871],{"style":6597},[35,9869],{"className":9870,"style":383},[382],[35,9872,9874],{"className":9873},[387,388,389,390],[35,9875,139],{"className":9876},[78,79,390],[35,9878,2258],{"className":9879},[2257],[35,9881,9883],{"className":9882},[371],[35,9884,9886],{"className":9885,"style":2265},[375],[35,9887],{},[35,9889,128],{"className":9890},[78],[35,9892],{"className":9893,"style":166},[165],[35,9895,136],{"className":9896},[170],[35,9898],{"className":9899,"style":166},[165],[35,9901,9903,9906,9909,9912,9920,9923,9926,9929],{"className":9902},[69],[35,9904],{"className":9905,"style":152},[73],[35,9907,238],{"className":9908},[278],[35,9910,7107],{"className":9911,"style":111},[78,79],[35,9913,9916],{"className":9914},[78,9915],"text",[35,9917,9819],{"className":9918},[78,9919],"cjk_fallback",[35,9921,248],{"className":9922},[296],[35,9924],{"className":9925,"style":881},[165],[35,9927,1361],{"className":9928},[715],[35,9930],{"className":9931,"style":881},[165],[35,9933,9935,9938,9941,9944,9950],{"className":9934},[69],[35,9936],{"className":9937,"style":152},[73],[35,9939,238],{"className":9940},[278],[35,9942,7107],{"className":9943,"style":111},[78,79],[35,9945,9947],{"className":9946},[78,9915],[35,9948,9830],{"className":9949},[78,9919],[35,9951,248],{"className":9952},[296],[464,9954,9955,9956],{},"代入数值：\n",[35,9957,9959,10011],{"className":9958},[38],[35,9960,9962],{"className":9961},[42],[44,9963,9964],{"xmlns":46},[48,9965,9966,10008],{},[51,9967,9968,9970,9972,9974,9976,9978,9980,9982,9984,9986,9988,9990,10000,10002,10004,10006],{},[54,9969,139],{},[134,9971,8820],{"stretchy":237},[134,9973,136],{},[134,9975,238],{"stretchy":237},[54,9977,139],{},[134,9979,591],{},[252,9981,254],{},[134,9983,248],{"stretchy":237},[134,9985,8820],{"stretchy":237},[134,9987,1361],{},[54,9989,128],{"mathvariant":127},[2203,9991,9992,9994],{},[54,9993,8643],{},[2203,9995,9996,9998],{},[54,9997,4621],{},[54,9999,139],{},[134,10001,238],{"stretchy":237},[54,10003,7107],{},[134,10005,248],{"stretchy":237},[54,10007,128],{"mathvariant":127},[58,10009,10010],{"encoding":60},"n! = (n-1)! \\cdot |C_{S_n}(\\sigma)|",[35,10012,10014,10035,10056,10077],{"className":10013,"ariaHidden":65},[64],[35,10015,10017,10020,10023,10026,10029,10032],{"className":10016},[69],[35,10018],{"className":10019,"style":211},[73],[35,10021,139],{"className":10022},[78,79],[35,10024,8820],{"className":10025},[296],[35,10027],{"className":10028,"style":166},[165],[35,10030,136],{"className":10031},[170],[35,10033],{"className":10034,"style":166},[165],[35,10036,10038,10041,10044,10047,10050,10053],{"className":10037},[69],[35,10039],{"className":10040,"style":152},[73],[35,10042,238],{"className":10043},[278],[35,10045,139],{"className":10046},[78,79],[35,10048],{"className":10049,"style":881},[165],[35,10051,591],{"className":10052},[715],[35,10054],{"className":10055,"style":881},[165],[35,10057,10059,10062,10065,10068,10071,10074],{"className":10058},[69],[35,10060],{"className":10061,"style":152},[73],[35,10063,254],{"className":10064},[78],[35,10066,8860],{"className":10067},[296],[35,10069],{"className":10070,"style":881},[165],[35,10072,1361],{"className":10073},[715],[35,10075],{"className":10076,"style":881},[165],[35,10078,10080,10083,10086,10166,10169,10172,10175],{"className":10079},[69],[35,10081],{"className":10082,"style":8676},[73],[35,10084,128],{"className":10085},[78],[35,10087,10089,10092],{"className":10088},[78],[35,10090,8643],{"className":10091,"style":8683},[78,79],[35,10093,10095],{"className":10094},[363],[35,10096,10098,10158],{"className":10097},[367,2234],[35,10099,10101,10155],{"className":10100},[371],[35,10102,10104],{"className":10103,"style":8696},[375],[35,10105,10106,10109],{"style":8699},[35,10107],{"className":10108,"style":383},[382],[35,10110,10112],{"className":10111},[387,388,389,390],[35,10113,10115],{"className":10114},[78,390],[35,10116,10118,10121],{"className":10117},[78,390],[35,10119,4621],{"className":10120,"style":4636},[78,79,390],[35,10122,10124],{"className":10123},[363],[35,10125,10127,10147],{"className":10126},[367,2234],[35,10128,10130,10144],{"className":10129},[371],[35,10131,10133],{"className":10132,"style":4502},[375],[35,10134,10135,10138],{"style":8729},[35,10136],{"className":10137,"style":4188},[382],[35,10139,10141],{"className":10140},[387,4192,4193,390],[35,10142,139],{"className":10143},[78,79,390],[35,10145,2258],{"className":10146},[2257],[35,10148,10150],{"className":10149},[371],[35,10151,10153],{"className":10152,"style":4206},[375],[35,10154],{},[35,10156,2258],{"className":10157},[2257],[35,10159,10161],{"className":10160},[371],[35,10162,10164],{"className":10163,"style":8759},[375],[35,10165],{},[35,10167,238],{"className":10168},[278],[35,10170,7107],{"className":10171,"style":111},[78,79],[35,10173,248],{"className":10174},[296],[35,10176,128],{"className":10177},[78],[464,10179,10180,10181,722],{},"解得：",[35,10182,10184,10244],{"className":10183},[38],[35,10185,10187],{"className":10186},[42],[44,10188,10189],{"xmlns":46},[48,10190,10191,10241],{},[51,10192,10193,10195,10205,10207,10209,10211,10213,10215,10237,10239],{},[54,10194,128],{"mathvariant":127},[2203,10196,10197,10199],{},[54,10198,8643],{},[2203,10200,10201,10203],{},[54,10202,4621],{},[54,10204,139],{},[134,10206,238],{"stretchy":237},[54,10208,7107],{},[134,10210,248],{"stretchy":237},[54,10212,128],{"mathvariant":127},[134,10214,136],{},[4041,10216,10217,10223],{},[51,10218,10219,10221],{},[54,10220,139],{},[134,10222,8820],{"stretchy":237},[51,10224,10225,10227,10229,10231,10233,10235],{},[134,10226,238],{"stretchy":237},[54,10228,139],{},[134,10230,591],{},[252,10232,254],{},[134,10234,248],{"stretchy":237},[134,10236,8820],{"stretchy":237},[134,10238,136],{},[54,10240,139],{},[58,10242,10243],{"encoding":60},"|C_{S_n}(\\sigma)| = \\frac{n!}{(n-1)!} = n",[35,10245,10247,10357,10457],{"className":10246,"ariaHidden":65},[64],[35,10248,10250,10253,10256,10336,10339,10342,10345,10348,10351,10354],{"className":10249},[69],[35,10251],{"className":10252,"style":8676},[73],[35,10254,128],{"className":10255},[78],[35,10257,10259,10262],{"className":10258},[78],[35,10260,8643],{"className":10261,"style":8683},[78,79],[35,10263,10265],{"className":10264},[363],[35,10266,10268,10328],{"className":10267},[367,2234],[35,10269,10271,10325],{"className":10270},[371],[35,10272,10274],{"className":10273,"style":8696},[375],[35,10275,10276,10279],{"style":8699},[35,10277],{"className":10278,"style":383},[382],[35,10280,10282],{"className":10281},[387,388,389,390],[35,10283,10285],{"className":10284},[78,390],[35,10286,10288,10291],{"className":10287},[78,390],[35,10289,4621],{"className":10290,"style":4636},[78,79,390],[35,10292,10294],{"className":10293},[363],[35,10295,10297,10317],{"className":10296},[367,2234],[35,10298,10300,10314],{"className":10299},[371],[35,10301,10303],{"className":10302,"style":4502},[375],[35,10304,10305,10308],{"style":8729},[35,10306],{"className":10307,"style":4188},[382],[35,10309,10311],{"className":10310},[387,4192,4193,390],[35,10312,139],{"className":10313},[78,79,390],[35,10315,2258],{"className":10316},[2257],[35,10318,10320],{"className":10319},[371],[35,10321,10323],{"className":10322,"style":4206},[375],[35,10324],{},[35,10326,2258],{"className":10327},[2257],[35,10329,10331],{"className":10330},[371],[35,10332,10334],{"className":10333,"style":8759},[375],[35,10335],{},[35,10337,238],{"className":10338},[278],[35,10340,7107],{"className":10341,"style":111},[78,79],[35,10343,248],{"className":10344},[296],[35,10346,128],{"className":10347},[78],[35,10349],{"className":10350,"style":166},[165],[35,10352,136],{"className":10353},[170],[35,10355],{"className":10356,"style":166},[165],[35,10358,10360,10364,10448,10451,10454],{"className":10359},[69],[35,10361],{"className":10362,"style":10363},[73],"height:1.4001em;vertical-align:-0.52em;",[35,10365,10367,10370,10445],{"className":10366},[78],[35,10368],{"className":10369},[278,4136],[35,10371,10373],{"className":10372},[4041],[35,10374,10376,10436],{"className":10375},[367,2234],[35,10377,10379,10433],{"className":10378},[371],[35,10380,10382,10408,10416],{"className":10381,"style":9520},[375],[35,10383,10384,10387],{"style":4152},[35,10385],{"className":10386,"style":4156},[382],[35,10388,10390],{"className":10389},[387,388,389,390],[35,10391,10393,10396,10399,10402,10405],{"className":10392},[78,390],[35,10394,238],{"className":10395},[278,390],[35,10397,139],{"className":10398},[78,79,390],[35,10400,591],{"className":10401},[715,390],[35,10403,254],{"className":10404},[78,390],[35,10406,8860],{"className":10407},[296,390],[35,10409,10410,10413],{"style":4211},[35,10411],{"className":10412,"style":4156},[382],[35,10414],{"className":10415,"style":4219},[4218],[35,10417,10418,10421],{"style":6303},[35,10419],{"className":10420,"style":4156},[382],[35,10422,10424],{"className":10423},[387,388,389,390],[35,10425,10427,10430],{"className":10426},[78,390],[35,10428,139],{"className":10429},[78,79,390],[35,10431,8820],{"className":10432},[296,390],[35,10434,2258],{"className":10435},[2257],[35,10437,10439],{"className":10438},[371],[35,10440,10443],{"className":10441,"style":10442},[375],"height:0.52em;",[35,10444],{},[35,10446],{"className":10447},[296,4136],[35,10449],{"className":10450,"style":166},[165],[35,10452,136],{"className":10453},[170],[35,10455],{"className":10456,"style":166},[165],[35,10458,10460,10463],{"className":10459},[69],[35,10461],{"className":10462,"style":180},[73],[35,10464,139],{"className":10465},[78,79],[464,10467,10468,469,10471],{},[30,10469,10470],{},"确定中心化子",[471,10472,10473,10707,10777],{},[464,10474,10475,10476,10677,10678,10706],{},"我们知道 ",[35,10477,10479,10535],{"className":10478},[38],[35,10480,10482],{"className":10481},[42],[44,10483,10484],{"xmlns":46},[48,10485,10486,10532],{},[51,10487,10488,10490,10492,10494,10496,10498,10500,10502,10504,10506,10512,10514,10516,10518,10530],{},[134,10489,427],{"stretchy":237},[54,10491,7107],{},[134,10493,432],{"stretchy":237},[134,10495,136],{},[134,10497,557],{"stretchy":237},[54,10499,560],{},[134,10501,243],{"separator":65},[54,10503,7107],{},[134,10505,243],{"separator":65},[330,10507,10508,10510],{},[54,10509,7107],{},[252,10511,573],{},[134,10513,243],{"separator":65},[134,10515,578],{},[134,10517,243],{"separator":65},[330,10519,10520,10522],{},[54,10521,7107],{},[51,10523,10524,10526,10528],{},[54,10525,139],{},[134,10527,591],{},[252,10529,254],{},[134,10531,596],{"stretchy":237},[58,10533,10534],{"encoding":60},"\\langle \\sigma \\rangle = \\{e, \\sigma, \\sigma^2, \\dots, \\sigma^{n-1}\\}",[35,10536,10538,10562],{"className":10537,"ariaHidden":65},[64],[35,10539,10541,10544,10547,10550,10553,10556,10559],{"className":10540},[69],[35,10542],{"className":10543,"style":152},[73],[35,10545,427],{"className":10546},[278],[35,10548,7107],{"className":10549,"style":111},[78,79],[35,10551,432],{"className":10552},[296],[35,10554],{"className":10555,"style":166},[165],[35,10557,136],{"className":10558},[170],[35,10560],{"className":10561,"style":166},[165],[35,10563,10565,10568,10571,10574,10577,10580,10583,10586,10589,10618,10621,10624,10627,10630,10633,10636,10674],{"className":10564},[69],[35,10566],{"className":10567,"style":609},[73],[35,10569,557],{"className":10570},[278],[35,10572,560],{"className":10573},[78,79],[35,10575,243],{"className":10576},[285],[35,10578],{"className":10579,"style":289},[165],[35,10581,7107],{"className":10582,"style":111},[78,79],[35,10584,243],{"className":10585},[285],[35,10587],{"className":10588,"style":289},[165],[35,10590,10592,10595],{"className":10591},[78],[35,10593,7107],{"className":10594,"style":111},[78,79],[35,10596,10598],{"className":10597},[363],[35,10599,10601],{"className":10600},[367],[35,10602,10604],{"className":10603},[371],[35,10605,10607],{"className":10606,"style":649},[375],[35,10608,10609,10612],{"style":378},[35,10610],{"className":10611,"style":383},[382],[35,10613,10615],{"className":10614},[387,388,389,390],[35,10616,573],{"className":10617},[78,390],[35,10619,243],{"className":10620},[285],[35,10622],{"className":10623,"style":289},[165],[35,10625,578],{"className":10626},[670],[35,10628],{"className":10629,"style":289},[165],[35,10631,243],{"className":10632},[285],[35,10634],{"className":10635,"style":289},[165],[35,10637,10639,10642],{"className":10638},[78],[35,10640,7107],{"className":10641,"style":111},[78,79],[35,10643,10645],{"className":10644},[363],[35,10646,10648],{"className":10647},[367],[35,10649,10651],{"className":10650},[371],[35,10652,10654],{"className":10653,"style":649},[375],[35,10655,10656,10659],{"style":378},[35,10657],{"className":10658,"style":383},[382],[35,10660,10662],{"className":10661},[387,388,389,390],[35,10663,10665,10668,10671],{"className":10664},[78,390],[35,10666,139],{"className":10667},[78,79,390],[35,10669,591],{"className":10670},[715,390],[35,10672,254],{"className":10673},[78,390],[35,10675,596],{"className":10676},[296]," 里面的元素肯定都和 ",[35,10679,10681,10694],{"className":10680},[38],[35,10682,10684],{"className":10683},[42],[44,10685,10686],{"xmlns":46},[48,10687,10688,10692],{},[51,10689,10690],{},[54,10691,7107],{},[58,10693,7514],{"encoding":60},[35,10695,10697],{"className":10696,"ariaHidden":65},[64],[35,10698,10700,10703],{"className":10699},[69],[35,10701],{"className":10702,"style":180},[73],[35,10704,7107],{"className":10705,"style":111},[78,79]," 交换（自己跟自己肯定交换）。",[464,10708,10709,10748,10749,722],{},[35,10710,10712,10730],{"className":10711},[38],[35,10713,10715],{"className":10714},[42],[44,10716,10717],{"xmlns":46},[48,10718,10719,10727],{},[51,10720,10721,10723,10725],{},[134,10722,427],{"stretchy":237},[54,10724,7107],{},[134,10726,432],{"stretchy":237},[58,10728,10729],{"encoding":60},"\\langle \\sigma \\rangle",[35,10731,10733],{"className":10732,"ariaHidden":65},[64],[35,10734,10736,10739,10742,10745],{"className":10735},[69],[35,10737],{"className":10738,"style":152},[73],[35,10740,427],{"className":10741},[278],[35,10743,7107],{"className":10744,"style":111},[78,79],[35,10746,432],{"className":10747},[296]," 的阶正好是 ",[35,10750,10752,10765],{"className":10751},[38],[35,10753,10755],{"className":10754},[42],[44,10756,10757],{"xmlns":46},[48,10758,10759,10763],{},[51,10760,10761],{},[54,10762,139],{},[58,10764,139],{"encoding":60},[35,10766,10768],{"className":10767,"ariaHidden":65},[64],[35,10769,10771,10774],{"className":10770},[69],[35,10772],{"className":10773,"style":180},[73],[35,10775,139],{"className":10776},[78,79],[464,10778,10779,10781,10782,10810,10811,10849,10850,722],{},[30,10780,3899],{},"：既然中心化子的大小是 ",[35,10783,10785,10798],{"className":10784},[38],[35,10786,10788],{"className":10787},[42],[44,10789,10790],{"xmlns":46},[48,10791,10792,10796],{},[51,10793,10794],{},[54,10795,139],{},[58,10797,139],{"encoding":60},[35,10799,10801],{"className":10800,"ariaHidden":65},[64],[35,10802,10804,10807],{"className":10803},[69],[35,10805],{"className":10806,"style":180},[73],[35,10808,139],{"className":10809},[78,79],"，而 ",[35,10812,10814,10831],{"className":10813},[38],[35,10815,10817],{"className":10816},[42],[44,10818,10819],{"xmlns":46},[48,10820,10821,10829],{},[51,10822,10823,10825,10827],{},[134,10824,427],{"stretchy":237},[54,10826,7107],{},[134,10828,432],{"stretchy":237},[58,10830,10729],{"encoding":60},[35,10832,10834],{"className":10833,"ariaHidden":65},[64],[35,10835,10837,10840,10843,10846],{"className":10836},[69],[35,10838],{"className":10839,"style":152},[73],[35,10841,427],{"className":10842},[278],[35,10844,7107],{"className":10845,"style":111},[78,79],[35,10847,432],{"className":10848},[296]," 是它在这个大小下的一个子集，那它们必须相等。即 ",[35,10851,10853,10888],{"className":10852},[38],[35,10854,10856],{"className":10855},[42],[44,10857,10858],{"xmlns":46},[48,10859,10860,10886],{},[51,10861,10862,10872,10874,10876,10878,10880,10882,10884],{},[2203,10863,10864,10866],{},[54,10865,8643],{},[2203,10867,10868,10870],{},[54,10869,4621],{},[54,10871,139],{},[134,10873,238],{"stretchy":237},[54,10875,7107],{},[134,10877,248],{"stretchy":237},[134,10879,136],{},[134,10881,427],{"stretchy":237},[54,10883,7107],{},[134,10885,432],{"stretchy":237},[58,10887,8666],{"encoding":60},[35,10889,10891,10995],{"className":10890,"ariaHidden":65},[64],[35,10892,10894,10897,10977,10980,10983,10986,10989,10992],{"className":10893},[69],[35,10895],{"className":10896,"style":8676},[73],[35,10898,10900,10903],{"className":10899},[78],[35,10901,8643],{"className":10902,"style":8683},[78,79],[35,10904,10906],{"className":10905},[363],[35,10907,10909,10969],{"className":10908},[367,2234],[35,10910,10912,10966],{"className":10911},[371],[35,10913,10915],{"className":10914,"style":8696},[375],[35,10916,10917,10920],{"style":8699},[35,10918],{"className":10919,"style":383},[382],[35,10921,10923],{"className":10922},[387,388,389,390],[35,10924,10926],{"className":10925},[78,390],[35,10927,10929,10932],{"className":10928},[78,390],[35,10930,4621],{"className":10931,"style":4636},[78,79,390],[35,10933,10935],{"className":10934},[363],[35,10936,10938,10958],{"className":10937},[367,2234],[35,10939,10941,10955],{"className":10940},[371],[35,10942,10944],{"className":10943,"style":4502},[375],[35,10945,10946,10949],{"style":8729},[35,10947],{"className":10948,"style":4188},[382],[35,10950,10952],{"className":10951},[387,4192,4193,390],[35,10953,139],{"className":10954},[78,79,390],[35,10956,2258],{"className":10957},[2257],[35,10959,10961],{"className":10960},[371],[35,10962,10964],{"className":10963,"style":4206},[375],[35,10965],{},[35,10967,2258],{"className":10968},[2257],[35,10970,10972],{"className":10971},[371],[35,10973,10975],{"className":10974,"style":8759},[375],[35,10976],{},[35,10978,238],{"className":10979},[278],[35,10981,7107],{"className":10982,"style":111},[78,79],[35,10984,248],{"className":10985},[296],[35,10987],{"className":10988,"style":166},[165],[35,10990,136],{"className":10991},[170],[35,10993],{"className":10994,"style":166},[165],[35,10996,10998,11001,11004,11007],{"className":10997},[69],[35,10999],{"className":11000,"style":152},[73],[35,11002,427],{"className":11003},[278],[35,11005,7107],{"className":11006,"style":111},[78,79],[35,11008,432],{"className":11009},[296],[14,11011],{},[22,11013,11015,11016,11085,11086,11140],{"id":11014},"题目-16sns_nsn-的正规子群n5n-ge-5n5","题目 16：",[35,11017,11019,11036],{"className":11018},[38],[35,11020,11022],{"className":11021},[42],[44,11023,11024],{"xmlns":46},[48,11025,11026,11034],{},[51,11027,11028],{},[2203,11029,11030,11032],{},[54,11031,4621],{},[54,11033,139],{},[58,11035,6566],{"encoding":60},[35,11037,11039],{"className":11038,"ariaHidden":65},[64],[35,11040,11042,11045],{"className":11041},[69],[35,11043],{"className":11044,"style":6576},[73],[35,11046,11048,11051],{"className":11047},[78],[35,11049,4621],{"className":11050,"style":4636},[78,79],[35,11052,11054],{"className":11053},[363],[35,11055,11057,11077],{"className":11056},[367,2234],[35,11058,11060,11074],{"className":11059},[371],[35,11061,11063],{"className":11062,"style":4756},[375],[35,11064,11065,11068],{"style":6597},[35,11066],{"className":11067,"style":383},[382],[35,11069,11071],{"className":11070},[387,388,389,390],[35,11072,139],{"className":11073},[78,79,390],[35,11075,2258],{"className":11076},[2257],[35,11078,11080],{"className":11079},[371],[35,11081,11083],{"className":11082,"style":2265},[375],[35,11084],{}," 的正规子群（",[35,11087,11089,11109],{"className":11088},[38],[35,11090,11092],{"className":11091},[42],[44,11093,11094],{"xmlns":46},[48,11095,11096,11106],{},[51,11097,11098,11100,11103],{},[54,11099,139],{},[134,11101,11102],{},"≥",[252,11104,11105],{},"5",[58,11107,11108],{"encoding":60},"n \\ge 5",[35,11110,11112,11131],{"className":11111,"ariaHidden":65},[64],[35,11113,11115,11119,11122,11125,11128],{"className":11114},[69],[35,11116],{"className":11117,"style":11118},[73],"height:0.7719em;vertical-align:-0.136em;",[35,11120,139],{"className":11121},[78,79],[35,11123],{"className":11124,"style":166},[165],[35,11126,11102],{"className":11127},[170],[35,11129],{"className":11130,"style":166},[165],[35,11132,11134,11137],{"className":11133},[69],[35,11135],{"className":11136,"style":312},[73],[35,11138,11105],{"className":11139},[78],"）",[27,11142,11143,6972,11145,11214,11215,722],{},[30,11144,32],{},[35,11146,11148,11165],{"className":11147},[38],[35,11149,11151],{"className":11150},[42],[44,11152,11153],{"xmlns":46},[48,11154,11155,11163],{},[51,11156,11157],{},[2203,11158,11159,11161],{},[54,11160,4621],{},[54,11162,139],{},[58,11164,6566],{"encoding":60},[35,11166,11168],{"className":11167,"ariaHidden":65},[64],[35,11169,11171,11174],{"className":11170},[69],[35,11172],{"className":11173,"style":6576},[73],[35,11175,11177,11180],{"className":11176},[78],[35,11178,4621],{"className":11179,"style":4636},[78,79],[35,11181,11183],{"className":11182},[363],[35,11184,11186,11206],{"className":11185},[367,2234],[35,11187,11189,11203],{"className":11188},[371],[35,11190,11192],{"className":11191,"style":4756},[375],[35,11193,11194,11197],{"style":6597},[35,11195],{"className":11196,"style":383},[382],[35,11198,11200],{"className":11199},[387,388,389,390],[35,11201,139],{"className":11202},[78,79,390],[35,11204,2258],{"className":11205},[2257],[35,11207,11209],{"className":11208},[371],[35,11210,11212],{"className":11211,"style":2265},[375],[35,11213],{}," 只有一个非平凡正规子群 ",[35,11216,11218,11235],{"className":11217},[38],[35,11219,11221],{"className":11220},[42],[44,11222,11223],{"xmlns":46},[48,11224,11225,11233],{},[51,11226,11227],{},[2203,11228,11229,11231],{},[54,11230,6635],{},[54,11232,139],{},[58,11234,6640],{"encoding":60},[35,11236,11238],{"className":11237,"ariaHidden":65},[64],[35,11239,11241,11244],{"className":11240},[69],[35,11242],{"className":11243,"style":6576},[73],[35,11245,11247,11250],{"className":11246},[78],[35,11248,6635],{"className":11249},[78,79],[35,11251,11253],{"className":11252},[363],[35,11254,11256,11276],{"className":11255},[367,2234],[35,11257,11259,11273],{"className":11258},[371],[35,11260,11262],{"className":11261,"style":4756},[375],[35,11263,11264,11267],{"style":2244},[35,11265],{"className":11266,"style":383},[382],[35,11268,11270],{"className":11269},[387,388,389,390],[35,11271,139],{"className":11272},[78,79,390],[35,11274,2258],{"className":11275},[2257],[35,11277,11279],{"className":11278},[371],[35,11280,11282],{"className":11281,"style":2265},[375],[35,11283],{},[27,11285,11286],{},[30,11287,11288],{},"【证明逻辑重构】",[461,11290,11291,11723,11924,14466],{},[464,11292,11293,11296,11297,1542,11325,11394,11395,11498,11499,11624,11625,11653,11654,722],{},[30,11294,11295],{},"目标","：假设 ",[35,11298,11300,11313],{"className":11299},[38],[35,11301,11303],{"className":11302},[42],[44,11304,11305],{"xmlns":46},[48,11306,11307,11311],{},[51,11308,11309],{},[54,11310,4005],{},[58,11312,4005],{"encoding":60},[35,11314,11316],{"className":11315,"ariaHidden":65},[64],[35,11317,11319,11322],{"className":11318},[69],[35,11320],{"className":11321,"style":74},[73],[35,11323,4005],{"className":11324,"style":4020},[78,79],[35,11326,11328,11345],{"className":11327},[38],[35,11329,11331],{"className":11330},[42],[44,11332,11333],{"xmlns":46},[48,11334,11335,11343],{},[51,11336,11337],{},[2203,11338,11339,11341],{},[54,11340,4621],{},[54,11342,139],{},[58,11344,6566],{"encoding":60},[35,11346,11348],{"className":11347,"ariaHidden":65},[64],[35,11349,11351,11354],{"className":11350},[69],[35,11352],{"className":11353,"style":6576},[73],[35,11355,11357,11360],{"className":11356},[78],[35,11358,4621],{"className":11359,"style":4636},[78,79],[35,11361,11363],{"className":11362},[363],[35,11364,11366,11386],{"className":11365},[367,2234],[35,11367,11369,11383],{"className":11368},[371],[35,11370,11372],{"className":11371,"style":4756},[375],[35,11373,11374,11377],{"style":6597},[35,11375],{"className":11376,"style":383},[382],[35,11378,11380],{"className":11379},[387,388,389,390],[35,11381,139],{"className":11382},[78,79,390],[35,11384,2258],{"className":11385},[2257],[35,11387,11389],{"className":11388},[371],[35,11390,11392],{"className":11391,"style":2265},[375],[35,11393],{}," 的一个正规子群，且 ",[35,11396,11398,11421],{"className":11397},[38],[35,11399,11401],{"className":11400},[42],[44,11402,11403],{"xmlns":46},[48,11404,11405,11418],{},[51,11406,11407,11409,11412,11414,11416],{},[54,11408,4005],{},[134,11410,11411],{"mathvariant":127},"≠",[134,11413,557],{"stretchy":237},[54,11415,560],{},[134,11417,596],{"stretchy":237},[58,11419,11420],{"encoding":60},"H \\neq \\{e\\}",[35,11422,11424,11483],{"className":11423,"ariaHidden":65},[64],[35,11425,11427,11431,11434,11437,11480],{"className":11426},[69],[35,11428],{"className":11429,"style":11430},[73],"height:0.8889em;vertical-align:-0.1944em;",[35,11432,4005],{"className":11433,"style":4020},[78,79],[35,11435],{"className":11436,"style":166},[165],[35,11438,11440,11473,11477],{"className":11439},[170],[35,11441,11443],{"className":11442},[170],[35,11444,11447],{"className":11445},[78,11446],"vbox",[35,11448,11451],{"className":11449},[11450],"thinbox",[35,11452,11455,11458,11469],{"className":11453},[11454],"rlap",[35,11456],{"className":11457,"style":11430},[73],[35,11459,11462],{"className":11460},[11461],"inner",[35,11463,11465],{"className":11464},[78],[35,11466,11468],{"className":11467},[170],"",[35,11470],{"className":11471},[11472],"fix",[35,11474],{"className":11475},[165,11476],"nobreak",[35,11478,136],{"className":11479},[170],[35,11481],{"className":11482,"style":166},[165],[35,11484,11486,11489,11492,11495],{"className":11485},[69],[35,11487],{"className":11488,"style":152},[73],[35,11490,557],{"className":11491},[278],[35,11493,560],{"className":11494},[78,79],[35,11496,596],{"className":11497},[296],"，",[35,11500,11502,11524],{"className":11501},[38],[35,11503,11505],{"className":11504},[42],[44,11506,11507],{"xmlns":46},[48,11508,11509,11521],{},[51,11510,11511,11513,11515],{},[54,11512,4005],{},[134,11514,11411],{"mathvariant":127},[2203,11516,11517,11519],{},[54,11518,4621],{},[54,11520,139],{},[58,11522,11523],{"encoding":60},"H \\neq S_n",[35,11525,11527,11578],{"className":11526,"ariaHidden":65},[64],[35,11528,11530,11533,11536,11539,11575],{"className":11529},[69],[35,11531],{"className":11532,"style":11430},[73],[35,11534,4005],{"className":11535,"style":4020},[78,79],[35,11537],{"className":11538,"style":166},[165],[35,11540,11542,11569,11572],{"className":11541},[170],[35,11543,11545],{"className":11544},[170],[35,11546,11548],{"className":11547},[78,11446],[35,11549,11551],{"className":11550},[11450],[35,11552,11554,11557,11566],{"className":11553},[11454],[35,11555],{"className":11556,"style":11430},[73],[35,11558,11560],{"className":11559},[11461],[35,11561,11563],{"className":11562},[78],[35,11564,11468],{"className":11565},[170],[35,11567],{"className":11568},[11472],[35,11570],{"className":11571},[165,11476],[35,11573,136],{"className":11574},[170],[35,11576],{"className":11577,"style":166},[165],[35,11579,11581,11584],{"className":11580},[69],[35,11582],{"className":11583,"style":6576},[73],[35,11585,11587,11590],{"className":11586},[78],[35,11588,4621],{"className":11589,"style":4636},[78,79],[35,11591,11593],{"className":11592},[363],[35,11594,11596,11616],{"className":11595},[367,2234],[35,11597,11599,11613],{"className":11598},[371],[35,11600,11602],{"className":11601,"style":4756},[375],[35,11603,11604,11607],{"style":6597},[35,11605],{"className":11606,"style":383},[382],[35,11608,11610],{"className":11609},[387,388,389,390],[35,11611,139],{"className":11612},[78,79,390],[35,11614,2258],{"className":11615},[2257],[35,11617,11619],{"className":11618},[371],[35,11620,11622],{"className":11621,"style":2265},[375],[35,11623],{},"。我们要证明 ",[35,11626,11628,11641],{"className":11627},[38],[35,11629,11631],{"className":11630},[42],[44,11632,11633],{"xmlns":46},[48,11634,11635,11639],{},[51,11636,11637],{},[54,11638,4005],{},[58,11640,4005],{"encoding":60},[35,11642,11644],{"className":11643,"ariaHidden":65},[64],[35,11645,11647,11650],{"className":11646},[69],[35,11648],{"className":11649,"style":74},[73],[35,11651,4005],{"className":11652,"style":4020},[78,79]," 只能是 ",[35,11655,11657,11674],{"className":11656},[38],[35,11658,11660],{"className":11659},[42],[44,11661,11662],{"xmlns":46},[48,11663,11664,11672],{},[51,11665,11666],{},[2203,11667,11668,11670],{},[54,11669,6635],{},[54,11671,139],{},[58,11673,6640],{"encoding":60},[35,11675,11677],{"className":11676,"ariaHidden":65},[64],[35,11678,11680,11683],{"className":11679},[69],[35,11681],{"className":11682,"style":6576},[73],[35,11684,11686,11689],{"className":11685},[78],[35,11687,6635],{"className":11688},[78,79],[35,11690,11692],{"className":11691},[363],[35,11693,11695,11715],{"className":11694},[367,2234],[35,11696,11698,11712],{"className":11697},[371],[35,11699,11701],{"className":11700,"style":4756},[375],[35,11702,11703,11706],{"style":2244},[35,11704],{"className":11705,"style":383},[382],[35,11707,11709],{"className":11708},[387,388,389,390],[35,11710,139],{"className":11711},[78,79,390],[35,11713,2258],{"className":11714},[2257],[35,11716,11718],{"className":11717},[371],[35,11719,11721],{"className":11720,"style":2265},[375],[35,11722],{},[464,11724,11725,7497,11798],{},[30,11726,11727,11728,11797],{},"关键点：",[35,11729,11731,11748],{"className":11730},[38],[35,11732,11734],{"className":11733},[42],[44,11735,11736],{"xmlns":46},[48,11737,11738,11746],{},[51,11739,11740],{},[2203,11741,11742,11744],{},[54,11743,6635],{},[54,11745,139],{},[58,11747,6640],{"encoding":60},[35,11749,11751],{"className":11750,"ariaHidden":65},[64],[35,11752,11754,11757],{"className":11753},[69],[35,11755],{"className":11756,"style":6576},[73],[35,11758,11760,11763],{"className":11759},[78],[35,11761,6635],{"className":11762},[78,79],[35,11764,11766],{"className":11765},[363],[35,11767,11769,11789],{"className":11768},[367,2234],[35,11770,11772,11786],{"className":11771},[371],[35,11773,11775],{"className":11774,"style":4756},[375],[35,11776,11777,11780],{"style":2244},[35,11778],{"className":11779,"style":383},[382],[35,11781,11783],{"className":11782},[387,388,389,390],[35,11784,139],{"className":11785},[78,79,390],[35,11787,2258],{"className":11788},[2257],[35,11790,11792],{"className":11791},[371],[35,11793,11795],{"className":11794,"style":2265},[375],[35,11796],{}," 是单群（Simple Group）",[471,11799,11800],{},[464,11801,11802,11803,11853,11854,11923],{},"对于 ",[35,11804,11806,11823],{"className":11805},[38],[35,11807,11809],{"className":11808},[42],[44,11810,11811],{"xmlns":46},[48,11812,11813,11821],{},[51,11814,11815,11817,11819],{},[54,11816,139],{},[134,11818,11102],{},[252,11820,11105],{},[58,11822,11108],{"encoding":60},[35,11824,11826,11844],{"className":11825,"ariaHidden":65},[64],[35,11827,11829,11832,11835,11838,11841],{"className":11828},[69],[35,11830],{"className":11831,"style":11118},[73],[35,11833,139],{"className":11834},[78,79],[35,11836],{"className":11837,"style":166},[165],[35,11839,11102],{"className":11840},[170],[35,11842],{"className":11843,"style":166},[165],[35,11845,11847,11850],{"className":11846},[69],[35,11848],{"className":11849,"style":312},[73],[35,11851,11105],{"className":11852},[78],"， ",[35,11855,11857,11874],{"className":11856},[38],[35,11858,11860],{"className":11859},[42],[44,11861,11862],{"xmlns":46},[48,11863,11864,11872],{},[51,11865,11866],{},[2203,11867,11868,11870],{},[54,11869,6635],{},[54,11871,139],{},[58,11873,6640],{"encoding":60},[35,11875,11877],{"className":11876,"ariaHidden":65},[64],[35,11878,11880,11883],{"className":11879},[69],[35,11881],{"className":11882,"style":6576},[73],[35,11884,11886,11889],{"className":11885},[78],[35,11887,6635],{"className":11888},[78,79],[35,11890,11892],{"className":11891},[363],[35,11893,11895,11915],{"className":11894},[367,2234],[35,11896,11898,11912],{"className":11897},[371],[35,11899,11901],{"className":11900,"style":4756},[375],[35,11902,11903,11906],{"style":2244},[35,11904],{"className":11905,"style":383},[382],[35,11907,11909],{"className":11908},[387,388,389,390],[35,11910,139],{"className":11911},[78,79,390],[35,11913,2258],{"className":11914},[2257],[35,11916,11918],{"className":11917},[371],[35,11919,11921],{"className":11920,"style":2265},[375],[35,11922],{}," 没有非平凡的正规子群。这是已知的大定理。",[464,11925,11926,3985,12022],{},[30,11927,11928,11929],{},"分析 ",[35,11930,11932,11955],{"className":11931},[38],[35,11933,11935],{"className":11934},[42],[44,11936,11937],{"xmlns":46},[48,11938,11939,11952],{},[51,11940,11941,11943,11946],{},[54,11942,4005],{},[134,11944,11945],{},"∩",[2203,11947,11948,11950],{},[54,11949,6635],{},[54,11951,139],{},[58,11953,11954],{"encoding":60},"H \\cap A_n",[35,11956,11958,11976],{"className":11957,"ariaHidden":65},[64],[35,11959,11961,11964,11967,11970,11973],{"className":11960},[69],[35,11962],{"className":11963,"style":74},[73],[35,11965,4005],{"className":11966,"style":4020},[78,79],[35,11968],{"className":11969,"style":881},[165],[35,11971,11945],{"className":11972},[715],[35,11974],{"className":11975,"style":881},[165],[35,11977,11979,11982],{"className":11978},[69],[35,11980],{"className":11981,"style":6576},[73],[35,11983,11985,11988],{"className":11984},[78],[35,11986,6635],{"className":11987},[78,79],[35,11989,11991],{"className":11990},[363],[35,11992,11994,12014],{"className":11993},[367,2234],[35,11995,11997,12011],{"className":11996},[371],[35,11998,12000],{"className":11999,"style":4756},[375],[35,12001,12002,12005],{"style":2244},[35,12003],{"className":12004,"style":383},[382],[35,12006,12008],{"className":12007},[387,388,389,390],[35,12009,139],{"className":12010},[78,79,390],[35,12012,2258],{"className":12013},[2257],[35,12015,12017],{"className":12016},[371],[35,12018,12020],{"className":12019,"style":2265},[375],[35,12021],{},[471,12023,12024,12357],{},[464,12025,12026,12027,757,12055,12124,12125,12194,12195,12286,12287,12356],{},"因为 ",[35,12028,12030,12043],{"className":12029},[38],[35,12031,12033],{"className":12032},[42],[44,12034,12035],{"xmlns":46},[48,12036,12037,12041],{},[51,12038,12039],{},[54,12040,4005],{},[58,12042,4005],{"encoding":60},[35,12044,12046],{"className":12045,"ariaHidden":65},[64],[35,12047,12049,12052],{"className":12048},[69],[35,12050],{"className":12051,"style":74},[73],[35,12053,4005],{"className":12054,"style":4020},[78,79],[35,12056,12058,12075],{"className":12057},[38],[35,12059,12061],{"className":12060},[42],[44,12062,12063],{"xmlns":46},[48,12064,12065,12073],{},[51,12066,12067],{},[2203,12068,12069,12071],{},[54,12070,6635],{},[54,12072,139],{},[58,12074,6640],{"encoding":60},[35,12076,12078],{"className":12077,"ariaHidden":65},[64],[35,12079,12081,12084],{"className":12080},[69],[35,12082],{"className":12083,"style":6576},[73],[35,12085,12087,12090],{"className":12086},[78],[35,12088,6635],{"className":12089},[78,79],[35,12091,12093],{"className":12092},[363],[35,12094,12096,12116],{"className":12095},[367,2234],[35,12097,12099,12113],{"className":12098},[371],[35,12100,12102],{"className":12101,"style":4756},[375],[35,12103,12104,12107],{"style":2244},[35,12105],{"className":12106,"style":383},[382],[35,12108,12110],{"className":12109},[387,388,389,390],[35,12111,139],{"className":12112},[78,79,390],[35,12114,2258],{"className":12115},[2257],[35,12117,12119],{"className":12118},[371],[35,12120,12122],{"className":12121,"style":2265},[375],[35,12123],{}," 都是 ",[35,12126,12128,12145],{"className":12127},[38],[35,12129,12131],{"className":12130},[42],[44,12132,12133],{"xmlns":46},[48,12134,12135,12143],{},[51,12136,12137],{},[2203,12138,12139,12141],{},[54,12140,4621],{},[54,12142,139],{},[58,12144,6566],{"encoding":60},[35,12146,12148],{"className":12147,"ariaHidden":65},[64],[35,12149,12151,12154],{"className":12150},[69],[35,12152],{"className":12153,"style":6576},[73],[35,12155,12157,12160],{"className":12156},[78],[35,12158,4621],{"className":12159,"style":4636},[78,79],[35,12161,12163],{"className":12162},[363],[35,12164,12166,12186],{"className":12165},[367,2234],[35,12167,12169,12183],{"className":12168},[371],[35,12170,12172],{"className":12171,"style":4756},[375],[35,12173,12174,12177],{"style":6597},[35,12175],{"className":12176,"style":383},[382],[35,12178,12180],{"className":12179},[387,388,389,390],[35,12181,139],{"className":12182},[78,79,390],[35,12184,2258],{"className":12185},[2257],[35,12187,12189],{"className":12188},[371],[35,12190,12192],{"className":12191,"style":2265},[375],[35,12193],{}," 的正规子群，所以它们的交集 ",[35,12196,12198,12219],{"className":12197},[38],[35,12199,12201],{"className":12200},[42],[44,12202,12203],{"xmlns":46},[48,12204,12205,12217],{},[51,12206,12207,12209,12211],{},[54,12208,4005],{},[134,12210,11945],{},[2203,12212,12213,12215],{},[54,12214,6635],{},[54,12216,139],{},[58,12218,11954],{"encoding":60},[35,12220,12222,12240],{"className":12221,"ariaHidden":65},[64],[35,12223,12225,12228,12231,12234,12237],{"className":12224},[69],[35,12226],{"className":12227,"style":74},[73],[35,12229,4005],{"className":12230,"style":4020},[78,79],[35,12232],{"className":12233,"style":881},[165],[35,12235,11945],{"className":12236},[715],[35,12238],{"className":12239,"style":881},[165],[35,12241,12243,12246],{"className":12242},[69],[35,12244],{"className":12245,"style":6576},[73],[35,12247,12249,12252],{"className":12248},[78],[35,12250,6635],{"className":12251},[78,79],[35,12253,12255],{"className":12254},[363],[35,12256,12258,12278],{"className":12257},[367,2234],[35,12259,12261,12275],{"className":12260},[371],[35,12262,12264],{"className":12263,"style":4756},[375],[35,12265,12266,12269],{"style":2244},[35,12267],{"className":12268,"style":383},[382],[35,12270,12272],{"className":12271},[387,388,389,390],[35,12273,139],{"className":12274},[78,79,390],[35,12276,2258],{"className":12277},[2257],[35,12279,12281],{"className":12280},[371],[35,12282,12284],{"className":12283,"style":2265},[375],[35,12285],{}," 也是 ",[35,12288,12290,12307],{"className":12289},[38],[35,12291,12293],{"className":12292},[42],[44,12294,12295],{"xmlns":46},[48,12296,12297,12305],{},[51,12298,12299],{},[2203,12300,12301,12303],{},[54,12302,6635],{},[54,12304,139],{},[58,12306,6640],{"encoding":60},[35,12308,12310],{"className":12309,"ariaHidden":65},[64],[35,12311,12313,12316],{"className":12312},[69],[35,12314],{"className":12315,"style":6576},[73],[35,12317,12319,12322],{"className":12318},[78],[35,12320,6635],{"className":12321},[78,79],[35,12323,12325],{"className":12324},[363],[35,12326,12328,12348],{"className":12327},[367,2234],[35,12329,12331,12345],{"className":12330},[371],[35,12332,12334],{"className":12333,"style":4756},[375],[35,12335,12336,12339],{"style":2244},[35,12337],{"className":12338,"style":383},[382],[35,12340,12342],{"className":12341},[387,388,389,390],[35,12343,139],{"className":12344},[78,79,390],[35,12346,2258],{"className":12347},[2257],[35,12349,12351],{"className":12350},[371],[35,12352,12354],{"className":12353,"style":2265},[375],[35,12355],{}," 的正规子群。",[464,12358,12026,12359,12428,12429],{},[35,12360,12362,12379],{"className":12361},[38],[35,12363,12365],{"className":12364},[42],[44,12366,12367],{"xmlns":46},[48,12368,12369,12377],{},[51,12370,12371],{},[2203,12372,12373,12375],{},[54,12374,6635],{},[54,12376,139],{},[58,12378,6640],{"encoding":60},[35,12380,12382],{"className":12381,"ariaHidden":65},[64],[35,12383,12385,12388],{"className":12384},[69],[35,12386],{"className":12387,"style":6576},[73],[35,12389,12391,12394],{"className":12390},[78],[35,12392,6635],{"className":12393},[78,79],[35,12395,12397],{"className":12396},[363],[35,12398,12400,12420],{"className":12399},[367,2234],[35,12401,12403,12417],{"className":12402},[371],[35,12404,12406],{"className":12405,"style":4756},[375],[35,12407,12408,12411],{"style":2244},[35,12409],{"className":12410,"style":383},[382],[35,12412,12414],{"className":12413},[387,388,389,390],[35,12415,139],{"className":12416},[78,79,390],[35,12418,2258],{"className":12419},[2257],[35,12421,12423],{"className":12422},[371],[35,12424,12426],{"className":12425,"style":2265},[375],[35,12427],{}," 是单群，这个交集只有两种可能：\n",[471,12430,12431,13151],{},[464,12432,12433,469,12436,12591,12592,7497,12685],{},[30,12434,12435],{},"情况 A",[35,12437,12439,12469],{"className":12438},[38],[35,12440,12442],{"className":12441},[42],[44,12443,12444],{"xmlns":46},[48,12445,12446,12466],{},[51,12447,12448,12450,12452,12458,12460],{},[54,12449,4005],{},[134,12451,11945],{},[2203,12453,12454,12456],{},[54,12455,6635],{},[54,12457,139],{},[134,12459,136],{},[2203,12461,12462,12464],{},[54,12463,6635],{},[54,12465,139],{},[58,12467,12468],{"encoding":60},"H \\cap A_n = A_n",[35,12470,12472,12490,12545],{"className":12471,"ariaHidden":65},[64],[35,12473,12475,12478,12481,12484,12487],{"className":12474},[69],[35,12476],{"className":12477,"style":74},[73],[35,12479,4005],{"className":12480,"style":4020},[78,79],[35,12482],{"className":12483,"style":881},[165],[35,12485,11945],{"className":12486},[715],[35,12488],{"className":12489,"style":881},[165],[35,12491,12493,12496,12536,12539,12542],{"className":12492},[69],[35,12494],{"className":12495,"style":6576},[73],[35,12497,12499,12502],{"className":12498},[78],[35,12500,6635],{"className":12501},[78,79],[35,12503,12505],{"className":12504},[363],[35,12506,12508,12528],{"className":12507},[367,2234],[35,12509,12511,12525],{"className":12510},[371],[35,12512,12514],{"className":12513,"style":4756},[375],[35,12515,12516,12519],{"style":2244},[35,12517],{"className":12518,"style":383},[382],[35,12520,12522],{"className":12521},[387,388,389,390],[35,12523,139],{"className":12524},[78,79,390],[35,12526,2258],{"className":12527},[2257],[35,12529,12531],{"className":12530},[371],[35,12532,12534],{"className":12533,"style":2265},[375],[35,12535],{},[35,12537],{"className":12538,"style":166},[165],[35,12540,136],{"className":12541},[170],[35,12543],{"className":12544,"style":166},[165],[35,12546,12548,12551],{"className":12547},[69],[35,12549],{"className":12550,"style":6576},[73],[35,12552,12554,12557],{"className":12553},[78],[35,12555,6635],{"className":12556},[78,79],[35,12558,12560],{"className":12559},[363],[35,12561,12563,12583],{"className":12562},[367,2234],[35,12564,12566,12580],{"className":12565},[371],[35,12567,12569],{"className":12568,"style":4756},[375],[35,12570,12571,12574],{"style":2244},[35,12572],{"className":12573,"style":383},[382],[35,12575,12577],{"className":12576},[387,388,389,390],[35,12578,139],{"className":12579},[78,79,390],[35,12581,2258],{"className":12582},[2257],[35,12584,12586],{"className":12585},[371],[35,12587,12589],{"className":12588,"style":2265},[375],[35,12590],{},"。这意味着 ",[35,12593,12595,12618],{"className":12594},[38],[35,12596,12598],{"className":12597},[42],[44,12599,12600],{"xmlns":46},[48,12601,12602,12615],{},[51,12603,12604,12610,12613],{},[2203,12605,12606,12608],{},[54,12607,6635],{},[54,12609,139],{},[134,12611,12612],{},"⊆",[54,12614,4005],{},[58,12616,12617],{"encoding":60},"A_n \\subseteq H",[35,12619,12621,12676],{"className":12620,"ariaHidden":65},[64],[35,12622,12624,12627,12667,12670,12673],{"className":12623},[69],[35,12625],{"className":12626,"style":6576},[73],[35,12628,12630,12633],{"className":12629},[78],[35,12631,6635],{"className":12632},[78,79],[35,12634,12636],{"className":12635},[363],[35,12637,12639,12659],{"className":12638},[367,2234],[35,12640,12642,12656],{"className":12641},[371],[35,12643,12645],{"className":12644,"style":4756},[375],[35,12646,12647,12650],{"style":2244},[35,12648],{"className":12649,"style":383},[382],[35,12651,12653],{"className":12652},[387,388,389,390],[35,12654,139],{"className":12655},[78,79,390],[35,12657,2258],{"className":12658},[2257],[35,12660,12662],{"className":12661},[371],[35,12663,12665],{"className":12664,"style":2265},[375],[35,12666],{},[35,12668],{"className":12669,"style":166},[165],[35,12671,12612],{"className":12672},[170],[35,12674],{"className":12675,"style":166},[165],[35,12677,12679,12682],{"className":12678},[69],[35,12680],{"className":12681,"style":74},[73],[35,12683,4005],{"className":12684,"style":4020},[78,79],[471,12686,12687],{},[464,12688,12689,12758,12759,12828,12829,12857,12858,12927,12928,13052,13053,11653,13081,13150],{},[35,12690,12692,12709],{"className":12691},[38],[35,12693,12695],{"className":12694},[42],[44,12696,12697],{"xmlns":46},[48,12698,12699,12707],{},[51,12700,12701],{},[2203,12702,12703,12705],{},[54,12704,6635],{},[54,12706,139],{},[58,12708,6640],{"encoding":60},[35,12710,12712],{"className":12711,"ariaHidden":65},[64],[35,12713,12715,12718],{"className":12714},[69],[35,12716],{"className":12717,"style":6576},[73],[35,12719,12721,12724],{"className":12720},[78],[35,12722,6635],{"className":12723},[78,79],[35,12725,12727],{"className":12726},[363],[35,12728,12730,12750],{"className":12729},[367,2234],[35,12731,12733,12747],{"className":12732},[371],[35,12734,12736],{"className":12735,"style":4756},[375],[35,12737,12738,12741],{"style":2244},[35,12739],{"className":12740,"style":383},[382],[35,12742,12744],{"className":12743},[387,388,389,390],[35,12745,139],{"className":12746},[78,79,390],[35,12748,2258],{"className":12749},[2257],[35,12751,12753],{"className":12752},[371],[35,12754,12756],{"className":12755,"style":2265},[375],[35,12757],{}," 的指数是 2（它占了 ",[35,12760,12762,12779],{"className":12761},[38],[35,12763,12765],{"className":12764},[42],[44,12766,12767],{"xmlns":46},[48,12768,12769,12777],{},[51,12770,12771],{},[2203,12772,12773,12775],{},[54,12774,4621],{},[54,12776,139],{},[58,12778,6566],{"encoding":60},[35,12780,12782],{"className":12781,"ariaHidden":65},[64],[35,12783,12785,12788],{"className":12784},[69],[35,12786],{"className":12787,"style":6576},[73],[35,12789,12791,12794],{"className":12790},[78],[35,12792,4621],{"className":12793,"style":4636},[78,79],[35,12795,12797],{"className":12796},[363],[35,12798,12800,12820],{"className":12799},[367,2234],[35,12801,12803,12817],{"className":12802},[371],[35,12804,12806],{"className":12805,"style":4756},[375],[35,12807,12808,12811],{"style":6597},[35,12809],{"className":12810,"style":383},[382],[35,12812,12814],{"className":12813},[387,388,389,390],[35,12815,139],{"className":12816},[78,79,390],[35,12818,2258],{"className":12819},[2257],[35,12821,12823],{"className":12822},[371],[35,12824,12826],{"className":12825,"style":2265},[375],[35,12827],{}," 的一半）。如果 ",[35,12830,12832,12845],{"className":12831},[38],[35,12833,12835],{"className":12834},[42],[44,12836,12837],{"xmlns":46},[48,12838,12839,12843],{},[51,12840,12841],{},[54,12842,4005],{},[58,12844,4005],{"encoding":60},[35,12846,12848],{"className":12847,"ariaHidden":65},[64],[35,12849,12851,12854],{"className":12850},[69],[35,12852],{"className":12853,"style":74},[73],[35,12855,4005],{"className":12856,"style":4020},[78,79]," 包含 ",[35,12859,12861,12878],{"className":12860},[38],[35,12862,12864],{"className":12863},[42],[44,12865,12866],{"xmlns":46},[48,12867,12868,12876],{},[51,12869,12870],{},[2203,12871,12872,12874],{},[54,12873,6635],{},[54,12875,139],{},[58,12877,6640],{"encoding":60},[35,12879,12881],{"className":12880,"ariaHidden":65},[64],[35,12882,12884,12887],{"className":12883},[69],[35,12885],{"className":12886,"style":6576},[73],[35,12888,12890,12893],{"className":12889},[78],[35,12891,6635],{"className":12892},[78,79],[35,12894,12896],{"className":12895},[363],[35,12897,12899,12919],{"className":12898},[367,2234],[35,12900,12902,12916],{"className":12901},[371],[35,12903,12905],{"className":12904,"style":4756},[375],[35,12906,12907,12910],{"style":2244},[35,12908],{"className":12909,"style":383},[382],[35,12911,12913],{"className":12912},[387,388,389,390],[35,12914,139],{"className":12915},[78,79,390],[35,12917,2258],{"className":12918},[2257],[35,12920,12922],{"className":12921},[371],[35,12923,12925],{"className":12924,"style":2265},[375],[35,12926],{}," 且 ",[35,12929,12931,12952],{"className":12930},[38],[35,12932,12934],{"className":12933},[42],[44,12935,12936],{"xmlns":46},[48,12937,12938,12950],{},[51,12939,12940,12942,12944],{},[54,12941,4005],{},[134,12943,11411],{"mathvariant":127},[2203,12945,12946,12948],{},[54,12947,4621],{},[54,12949,139],{},[58,12951,11523],{"encoding":60},[35,12953,12955,13006],{"className":12954,"ariaHidden":65},[64],[35,12956,12958,12961,12964,12967,13003],{"className":12957},[69],[35,12959],{"className":12960,"style":11430},[73],[35,12962,4005],{"className":12963,"style":4020},[78,79],[35,12965],{"className":12966,"style":166},[165],[35,12968,12970,12997,13000],{"className":12969},[170],[35,12971,12973],{"className":12972},[170],[35,12974,12976],{"className":12975},[78,11446],[35,12977,12979],{"className":12978},[11450],[35,12980,12982,12985,12994],{"className":12981},[11454],[35,12983],{"className":12984,"style":11430},[73],[35,12986,12988],{"className":12987},[11461],[35,12989,12991],{"className":12990},[78],[35,12992,11468],{"className":12993},[170],[35,12995],{"className":12996},[11472],[35,12998],{"className":12999},[165,11476],[35,13001,136],{"className":13002},[170],[35,13004],{"className":13005,"style":166},[165],[35,13007,13009,13012],{"className":13008},[69],[35,13010],{"className":13011,"style":6576},[73],[35,13013,13015,13018],{"className":13014},[78],[35,13016,4621],{"className":13017,"style":4636},[78,79],[35,13019,13021],{"className":13020},[363],[35,13022,13024,13044],{"className":13023},[367,2234],[35,13025,13027,13041],{"className":13026},[371],[35,13028,13030],{"className":13029,"style":4756},[375],[35,13031,13032,13035],{"style":6597},[35,13033],{"className":13034,"style":383},[382],[35,13036,13038],{"className":13037},[387,388,389,390],[35,13039,139],{"className":13040},[78,79,390],[35,13042,2258],{"className":13043},[2257],[35,13045,13047],{"className":13046},[371],[35,13048,13050],{"className":13049,"style":2265},[375],[35,13051],{},"，那 ",[35,13054,13056,13069],{"className":13055},[38],[35,13057,13059],{"className":13058},[42],[44,13060,13061],{"xmlns":46},[48,13062,13063,13067],{},[51,13064,13065],{},[54,13066,4005],{},[58,13068,4005],{"encoding":60},[35,13070,13072],{"className":13071,"ariaHidden":65},[64],[35,13073,13075,13078],{"className":13074},[69],[35,13076],{"className":13077,"style":74},[73],[35,13079,4005],{"className":13080,"style":4020},[78,79],[35,13082,13084,13101],{"className":13083},[38],[35,13085,13087],{"className":13086},[42],[44,13088,13089],{"xmlns":46},[48,13090,13091,13099],{},[51,13092,13093],{},[2203,13094,13095,13097],{},[54,13096,6635],{},[54,13098,139],{},[58,13100,6640],{"encoding":60},[35,13102,13104],{"className":13103,"ariaHidden":65},[64],[35,13105,13107,13110],{"className":13106},[69],[35,13108],{"className":13109,"style":6576},[73],[35,13111,13113,13116],{"className":13112},[78],[35,13114,6635],{"className":13115},[78,79],[35,13117,13119],{"className":13118},[363],[35,13120,13122,13142],{"className":13121},[367,2234],[35,13123,13125,13139],{"className":13124},[371],[35,13126,13128],{"className":13127,"style":4756},[375],[35,13129,13130,13133],{"style":2244},[35,13131],{"className":13132,"style":383},[382],[35,13134,13136],{"className":13135},[387,388,389,390],[35,13137,139],{"className":13138},[78,79,390],[35,13140,2258],{"className":13141},[2257],[35,13143,13145],{"className":13144},[371],[35,13146,13148],{"className":13147,"style":2265},[375],[35,13149],{},"。得证。",[464,13152,13153,469,13156,7497,13280],{},[30,13154,13155],{},"情况 B",[35,13157,13159,13189],{"className":13158},[38],[35,13160,13162],{"className":13161},[42],[44,13163,13164],{"xmlns":46},[48,13165,13166,13186],{},[51,13167,13168,13170,13172,13178,13180,13182,13184],{},[54,13169,4005],{},[134,13171,11945],{},[2203,13173,13174,13176],{},[54,13175,6635],{},[54,13177,139],{},[134,13179,136],{},[134,13181,557],{"stretchy":237},[54,13183,560],{},[134,13185,596],{"stretchy":237},[58,13187,13188],{"encoding":60},"H \\cap A_n = \\{e\\}",[35,13190,13192,13210,13265],{"className":13191,"ariaHidden":65},[64],[35,13193,13195,13198,13201,13204,13207],{"className":13194},[69],[35,13196],{"className":13197,"style":74},[73],[35,13199,4005],{"className":13200,"style":4020},[78,79],[35,13202],{"className":13203,"style":881},[165],[35,13205,11945],{"className":13206},[715],[35,13208],{"className":13209,"style":881},[165],[35,13211,13213,13216,13256,13259,13262],{"className":13212},[69],[35,13214],{"className":13215,"style":6576},[73],[35,13217,13219,13222],{"className":13218},[78],[35,13220,6635],{"className":13221},[78,79],[35,13223,13225],{"className":13224},[363],[35,13226,13228,13248],{"className":13227},[367,2234],[35,13229,13231,13245],{"className":13230},[371],[35,13232,13234],{"className":13233,"style":4756},[375],[35,13235,13236,13239],{"style":2244},[35,13237],{"className":13238,"style":383},[382],[35,13240,13242],{"className":13241},[387,388,389,390],[35,13243,139],{"className":13244},[78,79,390],[35,13246,2258],{"className":13247},[2257],[35,13249,13251],{"className":13250},[371],[35,13252,13254],{"className":13253,"style":2265},[375],[35,13255],{},[35,13257],{"className":13258,"style":166},[165],[35,13260,136],{"className":13261},[170],[35,13263],{"className":13264,"style":166},[165],[35,13266,13268,13271,13274,13277],{"className":13267},[69],[35,13269],{"className":13270,"style":152},[73],[35,13272,557],{"className":13273},[278],[35,13275,560],{"className":13276},[78,79],[35,13278,596],{"className":13279},[296],[471,13281,13282,13730,13910,13953,14337,14463],{},[464,13283,13284,13285,722],{},"根据同态定理或乘积公式：",[35,13286,13288,13372],{"className":13287},[38],[35,13289,13291],{"className":13290},[42],[44,13292,13293],{"xmlns":46},[48,13294,13295,13369],{},[51,13296,13297,13299,13301,13307,13309,13311,13349,13351,13353,13355,13357,13359],{},[54,13298,128],{"mathvariant":127},[54,13300,4005],{},[2203,13302,13303,13305],{},[54,13304,6635],{},[54,13306,139],{},[54,13308,128],{"mathvariant":127},[134,13310,136],{},[4041,13312,13313,13333],{},[51,13314,13315,13317,13319,13321,13323,13325,13331],{},[54,13316,128],{"mathvariant":127},[54,13318,4005],{},[54,13320,128],{"mathvariant":127},[134,13322,1361],{},[54,13324,128],{"mathvariant":127},[2203,13326,13327,13329],{},[54,13328,6635],{},[54,13330,139],{},[54,13332,128],{"mathvariant":127},[51,13334,13335,13337,13339,13341,13347],{},[54,13336,128],{"mathvariant":127},[54,13338,4005],{},[134,13340,11945],{},[2203,13342,13343,13345],{},[54,13344,6635],{},[54,13346,139],{},[54,13348,128],{"mathvariant":127},[134,13350,136],{},[54,13352,128],{"mathvariant":127},[54,13354,4005],{},[54,13356,128],{"mathvariant":127},[134,13358,1361],{},[4041,13360,13361,13367],{},[51,13362,13363,13365],{},[54,13364,139],{},[134,13366,8820],{"stretchy":237},[252,13368,573],{},[58,13370,13371],{"encoding":60},"|H A_n| = \\frac{|H| \\cdot |A_n|}{|H \\cap A_n|} = |H| \\cdot \\frac{n!}{2}",[35,13373,13375,13439,13629,13653],{"className":13374,"ariaHidden":65},[64],[35,13376,13378,13381,13384,13387,13427,13430,13433,13436],{"className":13377},[69],[35,13379],{"className":13380,"style":152},[73],[35,13382,128],{"className":13383},[78],[35,13385,4005],{"className":13386,"style":4020},[78,79],[35,13388,13390,13393],{"className":13389},[78],[35,13391,6635],{"className":13392},[78,79],[35,13394,13396],{"className":13395},[363],[35,13397,13399,13419],{"className":13398},[367,2234],[35,13400,13402,13416],{"className":13401},[371],[35,13403,13405],{"className":13404,"style":4756},[375],[35,13406,13407,13410],{"style":2244},[35,13408],{"className":13409,"style":383},[382],[35,13411,13413],{"className":13412},[387,388,389,390],[35,13414,139],{"className":13415},[78,79,390],[35,13417,2258],{"className":13418},[2257],[35,13420,13422],{"className":13421},[371],[35,13423,13425],{"className":13424,"style":2265},[375],[35,13426],{},[35,13428,128],{"className":13429},[78],[35,13431],{"className":13432,"style":166},[165],[35,13434,136],{"className":13435},[170],[35,13437],{"className":13438,"style":166},[165],[35,13440,13442,13446,13620,13623,13626],{"className":13441},[69],[35,13443],{"className":13444,"style":13445},[73],"height:1.53em;vertical-align:-0.52em;",[35,13447,13449,13452,13617],{"className":13448},[78],[35,13450],{"className":13451},[278,4136],[35,13453,13455],{"className":13454},[4041],[35,13456,13458,13609],{"className":13457},[367,2234],[35,13459,13461,13606],{"className":13460},[371],[35,13462,13465,13528,13536],{"className":13463,"style":13464},[375],"height:1.01em;",[35,13466,13467,13470],{"style":4152},[35,13468],{"className":13469,"style":4156},[382],[35,13471,13473],{"className":13472},[387,388,389,390],[35,13474,13476,13479,13482,13485,13525],{"className":13475},[78,390],[35,13477,128],{"className":13478},[78,390],[35,13480,4005],{"className":13481,"style":4020},[78,79,390],[35,13483,11945],{"className":13484},[715,390],[35,13486,13488,13491],{"className":13487},[78,390],[35,13489,6635],{"className":13490},[78,79,390],[35,13492,13494],{"className":13493},[363],[35,13495,13497,13517],{"className":13496},[367,2234],[35,13498,13500,13514],{"className":13499},[371],[35,13501,13503],{"className":13502,"style":4502},[375],[35,13504,13505,13508],{"style":4184},[35,13506],{"className":13507,"style":4188},[382],[35,13509,13511],{"className":13510},[387,4192,4193,390],[35,13512,139],{"className":13513},[78,79,390],[35,13515,2258],{"className":13516},[2257],[35,13518,13520],{"className":13519},[371],[35,13521,13523],{"className":13522,"style":4206},[375],[35,13524],{},[35,13526,128],{"className":13527},[78,390],[35,13529,13530,13533],{"style":4211},[35,13531],{"className":13532,"style":4156},[382],[35,13534],{"className":13535,"style":4219},[4218],[35,13537,13539,13542],{"style":13538},"top:-3.485em;",[35,13540],{"className":13541,"style":4156},[382],[35,13543,13545],{"className":13544},[387,388,389,390],[35,13546,13548,13551,13554,13557,13560,13563,13603],{"className":13547},[78,390],[35,13549,128],{"className":13550},[78,390],[35,13552,4005],{"className":13553,"style":4020},[78,79,390],[35,13555,128],{"className":13556},[78,390],[35,13558,1361],{"className":13559},[715,390],[35,13561,128],{"className":13562},[78,390],[35,13564,13566,13569],{"className":13565},[78,390],[35,13567,6635],{"className":13568},[78,79,390],[35,13570,13572],{"className":13571},[363],[35,13573,13575,13595],{"className":13574},[367,2234],[35,13576,13578,13592],{"className":13577},[371],[35,13579,13581],{"className":13580,"style":4502},[375],[35,13582,13583,13586],{"style":4184},[35,13584],{"className":13585,"style":4188},[382],[35,13587,13589],{"className":13588},[387,4192,4193,390],[35,13590,139],{"className":13591},[78,79,390],[35,13593,2258],{"className":13594},[2257],[35,13596,13598],{"className":13597},[371],[35,13599,13601],{"className":13600,"style":4206},[375],[35,13602],{},[35,13604,128],{"className":13605},[78,390],[35,13607,2258],{"className":13608},[2257],[35,13610,13612],{"className":13611},[371],[35,13613,13615],{"className":13614,"style":10442},[375],[35,13616],{},[35,13618],{"className":13619},[296,4136],[35,13621],{"className":13622,"style":166},[165],[35,13624,136],{"className":13625},[170],[35,13627],{"className":13628,"style":166},[165],[35,13630,13632,13635,13638,13641,13644,13647,13650],{"className":13631},[69],[35,13633],{"className":13634,"style":152},[73],[35,13636,128],{"className":13637},[78],[35,13639,4005],{"className":13640,"style":4020},[78,79],[35,13642,128],{"className":13643},[78],[35,13645],{"className":13646,"style":881},[165],[35,13648,1361],{"className":13649},[715],[35,13651],{"className":13652,"style":881},[165],[35,13654,13656,13659],{"className":13655},[69],[35,13657],{"className":13658,"style":9501},[73],[35,13660,13662,13665,13727],{"className":13661},[78],[35,13663],{"className":13664},[278,4136],[35,13666,13668],{"className":13667},[4041],[35,13669,13671,13719],{"className":13670},[367,2234],[35,13672,13674,13716],{"className":13673},[371],[35,13675,13677,13691,13699],{"className":13676,"style":9520},[375],[35,13678,13679,13682],{"style":4152},[35,13680],{"className":13681,"style":4156},[382],[35,13683,13685],{"className":13684},[387,388,389,390],[35,13686,13688],{"className":13687},[78,390],[35,13689,573],{"className":13690},[78,390],[35,13692,13693,13696],{"style":4211},[35,13694],{"className":13695,"style":4156},[382],[35,13697],{"className":13698,"style":4219},[4218],[35,13700,13701,13704],{"style":6303},[35,13702],{"className":13703,"style":4156},[382],[35,13705,13707],{"className":13706},[387,388,389,390],[35,13708,13710,13713],{"className":13709},[78,390],[35,13711,139],{"className":13712},[78,79,390],[35,13714,8820],{"className":13715},[296,390],[35,13717,2258],{"className":13718},[2257],[35,13720,13722],{"className":13721},[371],[35,13723,13725],{"className":13724,"style":5107},[375],[35,13726],{},[35,13728],{"className":13729},[296,4136],[464,13731,12026,13732,1542,13807,13876,13877,722],{},[35,13733,13735,13755],{"className":13734},[38],[35,13736,13738],{"className":13737},[42],[44,13739,13740],{"xmlns":46},[48,13741,13742,13752],{},[51,13743,13744,13746],{},[54,13745,4005],{},[2203,13747,13748,13750],{},[54,13749,6635],{},[54,13751,139],{},[58,13753,13754],{"encoding":60},"H A_n",[35,13756,13758],{"className":13757,"ariaHidden":65},[64],[35,13759,13761,13764,13767],{"className":13760},[69],[35,13762],{"className":13763,"style":6576},[73],[35,13765,4005],{"className":13766,"style":4020},[78,79],[35,13768,13770,13773],{"className":13769},[78],[35,13771,6635],{"className":13772},[78,79],[35,13774,13776],{"className":13775},[363],[35,13777,13779,13799],{"className":13778},[367,2234],[35,13780,13782,13796],{"className":13781},[371],[35,13783,13785],{"className":13784,"style":4756},[375],[35,13786,13787,13790],{"style":2244},[35,13788],{"className":13789,"style":383},[382],[35,13791,13793],{"className":13792},[387,388,389,390],[35,13794,139],{"className":13795},[78,79,390],[35,13797,2258],{"className":13798},[2257],[35,13800,13802],{"className":13801},[371],[35,13803,13805],{"className":13804,"style":2265},[375],[35,13806],{},[35,13808,13810,13827],{"className":13809},[38],[35,13811,13813],{"className":13812},[42],[44,13814,13815],{"xmlns":46},[48,13816,13817,13825],{},[51,13818,13819],{},[2203,13820,13821,13823],{},[54,13822,4621],{},[54,13824,139],{},[58,13826,6566],{"encoding":60},[35,13828,13830],{"className":13829,"ariaHidden":65},[64],[35,13831,13833,13836],{"className":13832},[69],[35,13834],{"className":13835,"style":6576},[73],[35,13837,13839,13842],{"className":13838},[78],[35,13840,4621],{"className":13841,"style":4636},[78,79],[35,13843,13845],{"className":13844},[363],[35,13846,13848,13868],{"className":13847},[367,2234],[35,13849,13851,13865],{"className":13850},[371],[35,13852,13854],{"className":13853,"style":4756},[375],[35,13855,13856,13859],{"style":6597},[35,13857],{"className":13858,"style":383},[382],[35,13860,13862],{"className":13861},[387,388,389,390],[35,13863,139],{"className":13864},[78,79,390],[35,13866,2258],{"className":13867},[2257],[35,13869,13871],{"className":13870},[371],[35,13872,13874],{"className":13873,"style":2265},[375],[35,13875],{}," 的子群，它的大小不能超过 ",[35,13878,13880,13895],{"className":13879},[38],[35,13881,13883],{"className":13882},[42],[44,13884,13885],{"xmlns":46},[48,13886,13887,13893],{},[51,13888,13889,13891],{},[54,13890,139],{},[134,13892,8820],{"stretchy":237},[58,13894,9228],{"encoding":60},[35,13896,13898],{"className":13897,"ariaHidden":65},[64],[35,13899,13901,13904,13907],{"className":13900},[69],[35,13902],{"className":13903,"style":211},[73],[35,13905,139],{"className":13906},[78,79],[35,13908,8820],{"className":13909},[296],[464,13911,13912,13913,13952],{},"这强迫 ",[35,13914,13916,13934],{"className":13915},[38],[35,13917,13919],{"className":13918},[42],[44,13920,13921],{"xmlns":46},[48,13922,13923,13931],{},[51,13924,13925,13927,13929],{},[54,13926,128],{"mathvariant":127},[54,13928,4005],{},[54,13930,128],{"mathvariant":127},[58,13932,13933],{"encoding":60},"|H|",[35,13935,13937],{"className":13936,"ariaHidden":65},[64],[35,13938,13940,13943,13946,13949],{"className":13939},[69],[35,13941],{"className":13942,"style":152},[73],[35,13944,128],{"className":13945},[78],[35,13947,4005],{"className":13948,"style":4020},[78,79],[35,13950,128],{"className":13951},[78]," 只能等于 2。",[464,13954,13955,13956,14017,14018,14092,14093,14121,14122,14151,14152,14221,14222,14250,14251,14336],{},"如果 ",[35,13957,13959,13981],{"className":13958},[38],[35,13960,13962],{"className":13961},[42],[44,13963,13964],{"xmlns":46},[48,13965,13966,13978],{},[51,13967,13968,13970,13972,13974,13976],{},[54,13969,128],{"mathvariant":127},[54,13971,4005],{},[54,13973,128],{"mathvariant":127},[134,13975,136],{},[252,13977,573],{},[58,13979,13980],{"encoding":60},"|H|=2",[35,13982,13984,14008],{"className":13983,"ariaHidden":65},[64],[35,13985,13987,13990,13993,13996,13999,14002,14005],{"className":13986},[69],[35,13988],{"className":13989,"style":152},[73],[35,13991,128],{"className":13992},[78],[35,13994,4005],{"className":13995,"style":4020},[78,79],[35,13997,128],{"className":13998},[78],[35,14000],{"className":14001,"style":166},[165],[35,14003,136],{"className":14004},[170],[35,14006],{"className":14007,"style":166},[165],[35,14009,14011,14014],{"className":14010},[69],[35,14012],{"className":14013,"style":312},[73],[35,14015,573],{"className":14016},[78],"，设 ",[35,14019,14021,14047],{"className":14020},[38],[35,14022,14024],{"className":14023},[42],[44,14025,14026],{"xmlns":46},[48,14027,14028,14044],{},[51,14029,14030,14032,14034,14036,14038,14040,14042],{},[54,14031,4005],{},[134,14033,136],{},[134,14035,557],{"stretchy":237},[54,14037,560],{},[134,14039,243],{"separator":65},[54,14041,6711],{},[134,14043,596],{"stretchy":237},[58,14045,14046],{"encoding":60},"H = \\{e, \\tau\\}",[35,14048,14050,14068],{"className":14049,"ariaHidden":65},[64],[35,14051,14053,14056,14059,14062,14065],{"className":14052},[69],[35,14054],{"className":14055,"style":74},[73],[35,14057,4005],{"className":14058,"style":4020},[78,79],[35,14060],{"className":14061,"style":166},[165],[35,14063,136],{"className":14064},[170],[35,14066],{"className":14067,"style":166},[165],[35,14069,14071,14074,14077,14080,14083,14086,14089],{"className":14070},[69],[35,14072],{"className":14073,"style":152},[73],[35,14075,557],{"className":14076},[278],[35,14078,560],{"className":14079},[78,79],[35,14081,243],{"className":14082},[285],[35,14084],{"className":14085,"style":289},[165],[35,14087,6711],{"className":14088,"style":6756},[78,79],[35,14090,596],{"className":14091},[296],"。因为 ",[35,14094,14096,14109],{"className":14095},[38],[35,14097,14099],{"className":14098},[42],[44,14100,14101],{"xmlns":46},[48,14102,14103,14107],{},[51,14104,14105],{},[54,14106,4005],{},[58,14108,4005],{"encoding":60},[35,14110,14112],{"className":14111,"ariaHidden":65},[64],[35,14113,14115,14118],{"className":14114},[69],[35,14116],{"className":14117,"style":74},[73],[35,14119,4005],{"className":14120,"style":4020},[78,79]," 是正规子群，这意味着 ",[35,14123,14125,14139],{"className":14124},[38],[35,14126,14128],{"className":14127},[42],[44,14129,14130],{"xmlns":46},[48,14131,14132,14136],{},[51,14133,14134],{},[54,14135,6711],{},[58,14137,14138],{"encoding":60},"\\tau",[35,14140,14142],{"className":14141,"ariaHidden":65},[64],[35,14143,14145,14148],{"className":14144},[69],[35,14146],{"className":14147,"style":180},[73],[35,14149,6711],{"className":14150,"style":6756},[78,79]," 必须和 ",[35,14153,14155,14172],{"className":14154},[38],[35,14156,14158],{"className":14157},[42],[44,14159,14160],{"xmlns":46},[48,14161,14162,14170],{},[51,14163,14164],{},[2203,14165,14166,14168],{},[54,14167,4621],{},[54,14169,139],{},[58,14171,6566],{"encoding":60},[35,14173,14175],{"className":14174,"ariaHidden":65},[64],[35,14176,14178,14181],{"className":14177},[69],[35,14179],{"className":14180,"style":6576},[73],[35,14182,14184,14187],{"className":14183},[78],[35,14185,4621],{"className":14186,"style":4636},[78,79],[35,14188,14190],{"className":14189},[363],[35,14191,14193,14213],{"className":14192},[367,2234],[35,14194,14196,14210],{"className":14195},[371],[35,14197,14199],{"className":14198,"style":4756},[375],[35,14200,14201,14204],{"style":6597},[35,14202],{"className":14203,"style":383},[382],[35,14205,14207],{"className":14206},[387,388,389,390],[35,14208,139],{"className":14209},[78,79,390],[35,14211,2258],{"className":14212},[2257],[35,14214,14216],{"className":14215},[371],[35,14217,14219],{"className":14218,"style":2265},[375],[35,14220],{}," 中所有元素交换（即 ",[35,14223,14225,14238],{"className":14224},[38],[35,14226,14228],{"className":14227},[42],[44,14229,14230],{"xmlns":46},[48,14231,14232,14236],{},[51,14233,14234],{},[54,14235,6711],{},[58,14237,14138],{"encoding":60},[35,14239,14241],{"className":14240,"ariaHidden":65},[64],[35,14242,14244,14247],{"className":14243},[69],[35,14245],{"className":14246,"style":180},[73],[35,14248,6711],{"className":14249,"style":6756},[78,79]," 在中心 ",[35,14252,14254,14278],{"className":14253},[38],[35,14255,14257],{"className":14256},[42],[44,14258,14259],{"xmlns":46},[48,14260,14261,14275],{},[51,14262,14263,14265,14267,14273],{},[54,14264,6464],{},[134,14266,238],{"stretchy":237},[2203,14268,14269,14271],{},[54,14270,4621],{},[54,14272,139],{},[134,14274,248],{"stretchy":237},[58,14276,14277],{"encoding":60},"Z(S_n)",[35,14279,14281],{"className":14280,"ariaHidden":65},[64],[35,14282,14284,14287,14290,14293,14333],{"className":14283},[69],[35,14285],{"className":14286,"style":152},[73],[35,14288,6464],{"className":14289,"style":8683},[78,79],[35,14291,238],{"className":14292},[278],[35,14294,14296,14299],{"className":14295},[78],[35,14297,4621],{"className":14298,"style":4636},[78,79],[35,14300,14302],{"className":14301},[363],[35,14303,14305,14325],{"className":14304},[367,2234],[35,14306,14308,14322],{"className":14307},[371],[35,14309,14311],{"className":14310,"style":4756},[375],[35,14312,14313,14316],{"style":6597},[35,14314],{"className":14315,"style":383},[382],[35,14317,14319],{"className":14318},[387,388,389,390],[35,14320,139],{"className":14321},[78,79,390],[35,14323,2258],{"className":14324},[2257],[35,14326,14328],{"className":14327},[371],[35,14329,14331],{"className":14330,"style":2265},[375],[35,14332],{},[35,14334,248],{"className":14335},[296]," 里）。",[464,14338,14339,14340,11498,14391,722],{},"但是！对于 ",[35,14341,14343,14361],{"className":14342},[38],[35,14344,14346],{"className":14345},[42],[44,14347,14348],{"xmlns":46},[48,14349,14350,14358],{},[51,14351,14352,14354,14356],{},[54,14353,139],{},[134,14355,11102],{},[252,14357,7599],{},[58,14359,14360],{"encoding":60},"n \\ge 3",[35,14362,14364,14382],{"className":14363,"ariaHidden":65},[64],[35,14365,14367,14370,14373,14376,14379],{"className":14366},[69],[35,14368],{"className":14369,"style":11118},[73],[35,14371,139],{"className":14372},[78,79],[35,14374],{"className":14375,"style":166},[165],[35,14377,11102],{"className":14378},[170],[35,14380],{"className":14381,"style":166},[165],[35,14383,14385,14388],{"className":14384},[69],[35,14386],{"className":14387,"style":312},[73],[35,14389,7599],{"className":14390},[78],[30,14392,14393,14462],{},[35,14394,14396,14413],{"className":14395},[38],[35,14397,14399],{"className":14398},[42],[44,14400,14401],{"xmlns":46},[48,14402,14403,14411],{},[51,14404,14405],{},[2203,14406,14407,14409],{},[54,14408,4621],{},[54,14410,139],{},[58,14412,6566],{"encoding":60},[35,14414,14416],{"className":14415,"ariaHidden":65},[64],[35,14417,14419,14422],{"className":14418},[69],[35,14420],{"className":14421,"style":6576},[73],[35,14423,14425,14428],{"className":14424},[78],[35,14426,4621],{"className":14427,"style":4636},[78,79],[35,14429,14431],{"className":14430},[363],[35,14432,14434,14454],{"className":14433},[367,2234],[35,14435,14437,14451],{"className":14436},[371],[35,14438,14440],{"className":14439,"style":4756},[375],[35,14441,14442,14445],{"style":6597},[35,14443],{"className":14444,"style":383},[382],[35,14446,14448],{"className":14447},[387,388,389,390],[35,14449,139],{"className":14450},[78,79,390],[35,14452,2258],{"className":14453},[2257],[35,14455,14457],{"className":14456},[371],[35,14458,14460],{"className":14459,"style":2265},[375],[35,14461],{}," 的中心是平凡的（只有单位元）",[464,14464,14465],{},"矛盾！所以情况 B 不可能发生。",[464,14467,14468,14470,14471,722],{},[30,14469,3899],{},"：只能是情况 A，即 ",[35,14472,14474,14496],{"className":14473},[38],[35,14475,14477],{"className":14476},[42],[44,14478,14479],{"xmlns":46},[48,14480,14481,14493],{},[51,14482,14483,14485,14487],{},[54,14484,4005],{},[134,14486,136],{},[2203,14488,14489,14491],{},[54,14490,6635],{},[54,14492,139],{},[58,14494,14495],{"encoding":60},"H = A_n",[35,14497,14499,14517],{"className":14498,"ariaHidden":65},[64],[35,14500,14502,14505,14508,14511,14514],{"className":14501},[69],[35,14503],{"className":14504,"style":74},[73],[35,14506,4005],{"className":14507,"style":4020},[78,79],[35,14509],{"className":14510,"style":166},[165],[35,14512,136],{"className":14513},[170],[35,14515],{"className":14516,"style":166},[165],[35,14518,14520,14523],{"className":14519},[69],[35,14521],{"className":14522,"style":6576},[73],[35,14524,14526,14529],{"className":14525},[78],[35,14527,6635],{"className":14528},[78,79],[35,14530,14532],{"className":14531},[363],[35,14533,14535,14555],{"className":14534},[367,2234],[35,14536,14538,14552],{"className":14537},[371],[35,14539,14541],{"className":14540,"style":4756},[375],[35,14542,14543,14546],{"style":2244},[35,14544],{"className":14545,"style":383},[382],[35,14547,14549],{"className":14548},[387,388,389,390],[35,14550,139],{"className":14551},[78,79,390],[35,14553,2258],{"className":14554},[2257],[35,14556,14558],{"className":14557},[371],[35,14559,14561],{"className":14560,"style":2265},[375],[35,14562],{},[14,14564],{},[22,14566,14568],{"id":14567},"题目-20只有两个自同构的群","题目 20：只有两个自同构的群",[27,14570,14571,14573,14574,14660,14661,14689],{},[30,14572,32],{}," 如果 ",[35,14575,14577,14609],{"className":14576},[38],[35,14578,14580],{"className":14579},[42],[44,14581,14582],{"xmlns":46},[48,14583,14584,14606],{},[51,14585,14586,14588,14590,14592,14594,14596,14598,14600,14602,14604],{},[54,14587,128],{"mathvariant":127},[54,14589,6635],{},[54,14591,801],{},[54,14593,9672],{},[134,14595,238],{"stretchy":237},[54,14597,56],{},[134,14599,248],{"stretchy":237},[54,14601,128],{"mathvariant":127},[134,14603,136],{},[252,14605,573],{},[58,14607,14608],{"encoding":60},"|Aut(G)| = 2",[35,14610,14612,14651],{"className":14611,"ariaHidden":65},[64],[35,14613,14615,14618,14621,14624,14627,14630,14633,14636,14639,14642,14645,14648],{"className":14614},[69],[35,14616],{"className":14617,"style":152},[73],[35,14619,128],{"className":14620},[78],[35,14622,6635],{"className":14623},[78,79],[35,14625,801],{"className":14626},[78,79],[35,14628,9672],{"className":14629},[78,79],[35,14631,238],{"className":14632},[278],[35,14634,56],{"className":14635},[78,79],[35,14637,248],{"className":14638},[296],[35,14640,128],{"className":14641},[78],[35,14643],{"className":14644,"style":166},[165],[35,14646,136],{"className":14647},[170],[35,14649],{"className":14650,"style":166},[165],[35,14652,14654,14657],{"className":14653},[69],[35,14655],{"className":14656,"style":312},[73],[35,14658,573],{"className":14659},[78],"，证明 ",[35,14662,14664,14677],{"className":14663},[38],[35,14665,14667],{"className":14666},[42],[44,14668,14669],{"xmlns":46},[48,14670,14671,14675],{},[51,14672,14673],{},[54,14674,56],{},[58,14676,56],{"encoding":60},[35,14678,14680],{"className":14679,"ariaHidden":65},[64],[35,14681,14683,14686],{"className":14682},[69],[35,14684],{"className":14685,"style":74},[73],[35,14687,56],{"className":14688},[78,79]," 是交换群。",[27,14691,14692],{},[30,14693,459],{},[461,14695,14696,14731,15068,15203],{},[464,14697,14698,14701,14702,14730],{},[30,14699,14700],{},"什么是自同构？"," 群 ",[35,14703,14705,14718],{"className":14704},[38],[35,14706,14708],{"className":14707},[42],[44,14709,14710],{"xmlns":46},[48,14711,14712,14716],{},[51,14713,14714],{},[54,14715,56],{},[58,14717,56],{"encoding":60},[35,14719,14721],{"className":14720,"ariaHidden":65},[64],[35,14722,14724,14727],{"className":14723},[69],[35,14725],{"className":14726,"style":74},[73],[35,14728,56],{"className":14729},[78,79]," 到它自己的同构映射。",[464,14732,14733,14736],{},[30,14734,14735],{},"什么是内自同构（Inn）？",[471,14737,14738,14956],{},[464,14739,14740,14741,14794,14795,14955],{},"对任意 ",[35,14742,14744,14763],{"className":14743},[38],[35,14745,14747],{"className":14746},[42],[44,14748,14749],{"xmlns":46},[48,14750,14751,14760],{},[51,14752,14753,14755,14758],{},[54,14754,95],{},[134,14756,14757],{},"∈",[54,14759,56],{},[58,14761,14762],{"encoding":60},"g \\in G",[35,14764,14766,14785],{"className":14765,"ariaHidden":65},[64],[35,14767,14769,14773,14776,14779,14782],{"className":14768},[69],[35,14770],{"className":14771,"style":14772},[73],"height:0.7335em;vertical-align:-0.1944em;",[35,14774,95],{"className":14775,"style":111},[78,79],[35,14777],{"className":14778,"style":166},[165],[35,14780,14757],{"className":14781},[170],[35,14783],{"className":14784,"style":166},[165],[35,14786,14788,14791],{"className":14787},[69],[35,14789],{"className":14790,"style":74},[73],[35,14792,56],{"className":14793},[78,79],"，定义映射 ",[35,14796,14798,14839],{"className":14797},[38],[35,14799,14801],{"className":14800},[42],[44,14802,14803],{"xmlns":46},[48,14804,14805,14836],{},[51,14806,14807,14814,14816,14818,14820,14822,14824,14826],{},[2203,14808,14809,14812],{},[54,14810,14811],{},"ϕ",[54,14813,95],{},[134,14815,238],{"stretchy":237},[54,14817,334],{},[134,14819,248],{"stretchy":237},[134,14821,136],{},[54,14823,95],{},[54,14825,334],{},[330,14827,14828,14830],{},[54,14829,95],{},[51,14831,14832,14834],{},[134,14833,591],{},[252,14835,254],{},[58,14837,14838],{"encoding":60},"\\phi_g(x) = g x g^{-1}",[35,14840,14842,14908],{"className":14841,"ariaHidden":65},[64],[35,14843,14845,14849,14890,14893,14896,14899,14902,14905],{"className":14844},[69],[35,14846],{"className":14847,"style":14848},[73],"height:1.0361em;vertical-align:-0.2861em;",[35,14850,14852,14855],{"className":14851},[78],[35,14853,14811],{"className":14854},[78,79],[35,14856,14858],{"className":14857},[363],[35,14859,14861,14881],{"className":14860},[367,2234],[35,14862,14864,14878],{"className":14863},[371],[35,14865,14867],{"className":14866,"style":4756},[375],[35,14868,14869,14872],{"style":2244},[35,14870],{"className":14871,"style":383},[382],[35,14873,14875],{"className":14874},[387,388,389,390],[35,14876,95],{"className":14877,"style":111},[78,79,390],[35,14879,2258],{"className":14880},[2257],[35,14882,14884],{"className":14883},[371],[35,14885,14888],{"className":14886,"style":14887},[375],"height:0.2861em;",[35,14889],{},[35,14891,238],{"className":14892},[278],[35,14894,334],{"className":14895},[78,79],[35,14897,248],{"className":14898},[296],[35,14900],{"className":14901,"style":166},[165],[35,14903,136],{"className":14904},[170],[35,14906],{"className":14907,"style":166},[165],[35,14909,14911,14914,14917,14920],{"className":14910},[69],[35,14912],{"className":14913,"style":1238},[73],[35,14915,95],{"className":14916,"style":111},[78,79],[35,14918,334],{"className":14919},[78,79],[35,14921,14923,14926],{"className":14922},[78],[35,14924,95],{"className":14925,"style":111},[78,79],[35,14927,14929],{"className":14928},[363],[35,14930,14932],{"className":14931},[367],[35,14933,14935],{"className":14934},[371],[35,14936,14938],{"className":14937,"style":649},[375],[35,14939,14940,14943],{"style":378},[35,14941],{"className":14942,"style":383},[382],[35,14944,14946],{"className":14945},[387,388,389,390],[35,14947,14949,14952],{"className":14948},[78,390],[35,14950,591],{"className":14951},[78,390],[35,14953,254],{"className":14954},[78,390],"。这也是一种自同构。",[464,14957,14958,14959,1542,15013,15067],{},"所有的内自同构构成的群 ",[35,14960,14962,14987],{"className":14961},[38],[35,14963,14965],{"className":14964},[42],[44,14966,14967],{"xmlns":46},[48,14968,14969,14984],{},[51,14970,14971,14974,14976,14978,14980,14982],{},[54,14972,14973],{},"I",[54,14975,139],{},[54,14977,139],{},[134,14979,238],{"stretchy":237},[54,14981,56],{},[134,14983,248],{"stretchy":237},[58,14985,14986],{"encoding":60},"Inn(G)",[35,14988,14990],{"className":14989,"ariaHidden":65},[64],[35,14991,14993,14996,15000,15004,15007,15010],{"className":14992},[69],[35,14994],{"className":14995,"style":152},[73],[35,14997,14973],{"className":14998,"style":14999},[78,79],"margin-right:0.0785em;",[35,15001,15003],{"className":15002},[78,79],"nn",[35,15005,238],{"className":15006},[278],[35,15008,56],{"className":15009},[78,79],[35,15011,248],{"className":15012},[296],[35,15014,15016,15040],{"className":15015},[38],[35,15017,15019],{"className":15018},[42],[44,15020,15021],{"xmlns":46},[48,15022,15023,15037],{},[51,15024,15025,15027,15029,15031,15033,15035],{},[54,15026,6635],{},[54,15028,801],{},[54,15030,9672],{},[134,15032,238],{"stretchy":237},[54,15034,56],{},[134,15036,248],{"stretchy":237},[58,15038,15039],{"encoding":60},"Aut(G)",[35,15041,15043],{"className":15042,"ariaHidden":65},[64],[35,15044,15046,15049,15052,15055,15058,15061,15064],{"className":15045},[69],[35,15047],{"className":15048,"style":152},[73],[35,15050,6635],{"className":15051},[78,79],[35,15053,801],{"className":15054},[78,79],[35,15056,9672],{"className":15057},[78,79],[35,15059,238],{"className":15060},[278],[35,15062,56],{"className":15063},[78,79],[35,15065,248],{"className":15066},[296]," 的子群。",[464,15069,15070,3985,15073,15173,15174,15202],{},[30,15071,15072],{},"关键联系",[35,15074,15076,15116],{"className":15075},[38],[35,15077,15079],{"className":15078},[42],[44,15080,15081],{"xmlns":46},[48,15082,15083,15113],{},[51,15084,15085,15087,15089,15091,15093,15095,15097,15100,15102,15105,15107,15109,15111],{},[54,15086,14973],{},[54,15088,139],{},[54,15090,139],{},[134,15092,238],{"stretchy":237},[54,15094,56],{},[134,15096,248],{"stretchy":237},[134,15098,15099],{},"≅",[54,15101,56],{},[54,15103,15104],{"mathvariant":127},"\u002F",[54,15106,6464],{},[134,15108,238],{"stretchy":237},[54,15110,56],{},[134,15112,248],{"stretchy":237},[58,15114,15115],{"encoding":60},"Inn(G) \\cong G \u002F Z(G)",[35,15117,15119,15149],{"className":15118,"ariaHidden":65},[64],[35,15120,15122,15125,15128,15131,15134,15137,15140,15143,15146],{"className":15121},[69],[35,15123],{"className":15124,"style":152},[73],[35,15126,14973],{"className":15127,"style":14999},[78,79],[35,15129,15003],{"className":15130},[78,79],[35,15132,238],{"className":15133},[278],[35,15135,56],{"className":15136},[78,79],[35,15138,248],{"className":15139},[296],[35,15141],{"className":15142,"style":166},[165],[35,15144,15099],{"className":15145},[170],[35,15147],{"className":15148,"style":166},[165],[35,15150,15152,15155,15158,15161,15164,15167,15170],{"className":15151},[69],[35,15153],{"className":15154,"style":152},[73],[35,15156,56],{"className":15157},[78,79],[35,15159,15104],{"className":15160},[78],[35,15162,6464],{"className":15163,"style":8683},[78,79],[35,15165,238],{"className":15166},[278],[35,15168,56],{"className":15169},[78,79],[35,15171,248],{"className":15172},[296],"\n即：内自同构群同构于 ",[35,15175,15177,15190],{"className":15176},[38],[35,15178,15180],{"className":15179},[42],[44,15181,15182],{"xmlns":46},[48,15183,15184,15188],{},[51,15185,15186],{},[54,15187,56],{},[58,15189,56],{"encoding":60},[35,15191,15193],{"className":15192,"ariaHidden":65},[64],[35,15194,15196,15199],{"className":15195},[69],[35,15197],{"className":15198,"style":74},[73],[35,15200,56],{"className":15201},[78,79]," 商掉它的中心。",[464,15204,15205,3985,15208],{},[30,15206,15207],{},"推导",[471,15209,15210,15298,15451,15728],{},[464,15211,15212,15213,722],{},"题目说 ",[35,15214,15216,15247],{"className":15215},[38],[35,15217,15219],{"className":15218},[42],[44,15220,15221],{"xmlns":46},[48,15222,15223,15245],{},[51,15224,15225,15227,15229,15231,15233,15235,15237,15239,15241,15243],{},[54,15226,128],{"mathvariant":127},[54,15228,6635],{},[54,15230,801],{},[54,15232,9672],{},[134,15234,238],{"stretchy":237},[54,15236,56],{},[134,15238,248],{"stretchy":237},[54,15240,128],{"mathvariant":127},[134,15242,136],{},[252,15244,573],{},[58,15246,14608],{"encoding":60},[35,15248,15250,15289],{"className":15249,"ariaHidden":65},[64],[35,15251,15253,15256,15259,15262,15265,15268,15271,15274,15277,15280,15283,15286],{"className":15252},[69],[35,15254],{"className":15255,"style":152},[73],[35,15257,128],{"className":15258},[78],[35,15260,6635],{"className":15261},[78,79],[35,15263,801],{"className":15264},[78,79],[35,15266,9672],{"className":15267},[78,79],[35,15269,238],{"className":15270},[278],[35,15272,56],{"className":15273},[78,79],[35,15275,248],{"className":15276},[296],[35,15278,128],{"className":15279},[78],[35,15281],{"className":15282,"style":166},[165],[35,15284,136],{"className":15285},[170],[35,15287],{"className":15288,"style":166},[165],[35,15290,15292,15295],{"className":15291},[69],[35,15293],{"className":15294,"style":312},[73],[35,15296,573],{"className":15297},[78],[464,15299,12026,15300,15399,15400,15450],{},[35,15301,15303,15342],{"className":15302},[38],[35,15304,15306],{"className":15305},[42],[44,15307,15308],{"xmlns":46},[48,15309,15310,15339],{},[51,15311,15312,15314,15316,15318,15320,15322,15324,15327,15329,15331,15333,15335,15337],{},[54,15313,14973],{},[54,15315,139],{},[54,15317,139],{},[134,15319,238],{"stretchy":237},[54,15321,56],{},[134,15323,248],{"stretchy":237},[134,15325,15326],{},"≤",[54,15328,6635],{},[54,15330,801],{},[54,15332,9672],{},[134,15334,238],{"stretchy":237},[54,15336,56],{},[134,15338,248],{"stretchy":237},[58,15340,15341],{"encoding":60},"Inn(G) \\le Aut(G)",[35,15343,15345,15375],{"className":15344,"ariaHidden":65},[64],[35,15346,15348,15351,15354,15357,15360,15363,15366,15369,15372],{"className":15347},[69],[35,15349],{"className":15350,"style":152},[73],[35,15352,14973],{"className":15353,"style":14999},[78,79],[35,15355,15003],{"className":15356},[78,79],[35,15358,238],{"className":15359},[278],[35,15361,56],{"className":15362},[78,79],[35,15364,248],{"className":15365},[296],[35,15367],{"className":15368,"style":166},[165],[35,15370,15326],{"className":15371},[170],[35,15373],{"className":15374,"style":166},[165],[35,15376,15378,15381,15384,15387,15390,15393,15396],{"className":15377},[69],[35,15379],{"className":15380,"style":152},[73],[35,15382,6635],{"className":15383},[78,79],[35,15385,801],{"className":15386},[78,79],[35,15388,9672],{"className":15389},[78,79],[35,15391,238],{"className":15392},[278],[35,15394,56],{"className":15395},[78,79],[35,15397,248],{"className":15398},[296],"，根据拉格朗日定理，",[35,15401,15403,15426],{"className":15402},[38],[35,15404,15406],{"className":15405},[42],[44,15407,15408],{"xmlns":46},[48,15409,15410,15424],{},[51,15411,15412,15414,15416,15418,15420,15422],{},[54,15413,14973],{},[54,15415,139],{},[54,15417,139],{},[134,15419,238],{"stretchy":237},[54,15421,56],{},[134,15423,248],{"stretchy":237},[58,15425,14986],{"encoding":60},[35,15427,15429],{"className":15428,"ariaHidden":65},[64],[35,15430,15432,15435,15438,15441,15444,15447],{"className":15431},[69],[35,15433],{"className":15434,"style":152},[73],[35,15436,14973],{"className":15437,"style":14999},[78,79],[35,15439,15003],{"className":15440},[78,79],[35,15442,238],{"className":15443},[278],[35,15445,56],{"className":15446},[78,79],[35,15448,248],{"className":15449},[296]," 的阶只能是 1 或者 2。",[464,15452,15453,469,15456,7497,15539],{},[30,15454,15455],{},"情况 1",[35,15457,15459,15491],{"className":15458},[38],[35,15460,15462],{"className":15461},[42],[44,15463,15464],{"xmlns":46},[48,15465,15466,15488],{},[51,15467,15468,15470,15472,15474,15476,15478,15480,15482,15484,15486],{},[54,15469,128],{"mathvariant":127},[54,15471,14973],{},[54,15473,139],{},[54,15475,139],{},[134,15477,238],{"stretchy":237},[54,15479,56],{},[134,15481,248],{"stretchy":237},[54,15483,128],{"mathvariant":127},[134,15485,136],{},[252,15487,254],{},[58,15489,15490],{"encoding":60},"|Inn(G)| = 1",[35,15492,15494,15530],{"className":15493,"ariaHidden":65},[64],[35,15495,15497,15500,15503,15506,15509,15512,15515,15518,15521,15524,15527],{"className":15496},[69],[35,15498],{"className":15499,"style":152},[73],[35,15501,128],{"className":15502},[78],[35,15504,14973],{"className":15505,"style":14999},[78,79],[35,15507,15003],{"className":15508},[78,79],[35,15510,238],{"className":15511},[278],[35,15513,56],{"className":15514},[78,79],[35,15516,248],{"className":15517},[296],[35,15519,128],{"className":15520},[78],[35,15522],{"className":15523,"style":166},[165],[35,15525,136],{"className":15526},[170],[35,15528],{"className":15529,"style":166},[165],[35,15531,15533,15536],{"className":15532},[69],[35,15534],{"className":15535,"style":312},[73],[35,15537,254],{"className":15538},[78],[471,15540,15541,15665],{},[464,15542,15543,15544,15598,15599,722],{},"这意味着 ",[35,15545,15547,15571],{"className":15546},[38],[35,15548,15550],{"className":15549},[42],[44,15551,15552],{"xmlns":46},[48,15553,15554,15568],{},[51,15555,15556,15558,15560,15562,15564,15566],{},[54,15557,56],{},[54,15559,15104],{"mathvariant":127},[54,15561,6464],{},[134,15563,238],{"stretchy":237},[54,15565,56],{},[134,15567,248],{"stretchy":237},[58,15569,15570],{"encoding":60},"G\u002FZ(G)",[35,15572,15574],{"className":15573,"ariaHidden":65},[64],[35,15575,15577,15580,15583,15586,15589,15592,15595],{"className":15576},[69],[35,15578],{"className":15579,"style":152},[73],[35,15581,56],{"className":15582},[78,79],[35,15584,15104],{"className":15585},[78],[35,15587,6464],{"className":15588,"style":8683},[78,79],[35,15590,238],{"className":15591},[278],[35,15593,56],{"className":15594},[78,79],[35,15596,248],{"className":15597},[296]," 只有一个元素。说明 ",[35,15600,15602,15626],{"className":15601},[38],[35,15603,15605],{"className":15604},[42],[44,15606,15607],{"xmlns":46},[48,15608,15609,15623],{},[51,15610,15611,15613,15615,15617,15619,15621],{},[54,15612,56],{},[134,15614,136],{},[54,15616,6464],{},[134,15618,238],{"stretchy":237},[54,15620,56],{},[134,15622,248],{"stretchy":237},[58,15624,15625],{"encoding":60},"G = Z(G)",[35,15627,15629,15647],{"className":15628,"ariaHidden":65},[64],[35,15630,15632,15635,15638,15641,15644],{"className":15631},[69],[35,15633],{"className":15634,"style":74},[73],[35,15636,56],{"className":15637},[78,79],[35,15639],{"className":15640,"style":166},[165],[35,15642,136],{"className":15643},[170],[35,15645],{"className":15646,"style":166},[165],[35,15648,15650,15653,15656,15659,15662],{"className":15649},[69],[35,15651],{"className":15652,"style":152},[73],[35,15654,6464],{"className":15655,"style":8683},[78,79],[35,15657,238],{"className":15658},[278],[35,15660,56],{"className":15661},[78,79],[35,15663,248],{"className":15664},[296],[464,15666,15667,15668,15696,15697,722],{},"中心等于全集，说明 ",[35,15669,15671,15684],{"className":15670},[38],[35,15672,15674],{"className":15673},[42],[44,15675,15676],{"xmlns":46},[48,15677,15678,15682],{},[51,15679,15680],{},[54,15681,56],{},[58,15683,56],{"encoding":60},[35,15685,15687],{"className":15686,"ariaHidden":65},[64],[35,15688,15690,15693],{"className":15689},[69],[35,15691],{"className":15692,"style":74},[73],[35,15694,56],{"className":15695},[78,79]," 中所有元素都可交换。",[30,15698,15699,15727],{},[35,15700,15702,15715],{"className":15701},[38],[35,15703,15705],{"className":15704},[42],[44,15706,15707],{"xmlns":46},[48,15708,15709,15713],{},[51,15710,15711],{},[54,15712,56],{},[58,15714,56],{"encoding":60},[35,15716,15718],{"className":15717,"ariaHidden":65},[64],[35,15719,15721,15724],{"className":15720},[69],[35,15722],{"className":15723,"style":74},[73],[35,15725,56],{"className":15726},[78,79]," 是交换群",[464,15729,15730,469,15733,7497,15816],{},[30,15731,15732],{},"情况 2",[35,15734,15736,15768],{"className":15735},[38],[35,15737,15739],{"className":15738},[42],[44,15740,15741],{"xmlns":46},[48,15742,15743,15765],{},[51,15744,15745,15747,15749,15751,15753,15755,15757,15759,15761,15763],{},[54,15746,128],{"mathvariant":127},[54,15748,14973],{},[54,15750,139],{},[54,15752,139],{},[134,15754,238],{"stretchy":237},[54,15756,56],{},[134,15758,248],{"stretchy":237},[54,15760,128],{"mathvariant":127},[134,15762,136],{},[252,15764,573],{},[58,15766,15767],{"encoding":60},"|Inn(G)| = 2",[35,15769,15771,15807],{"className":15770,"ariaHidden":65},[64],[35,15772,15774,15777,15780,15783,15786,15789,15792,15795,15798,15801,15804],{"className":15773},[69],[35,15775],{"className":15776,"style":152},[73],[35,15778,128],{"className":15779},[78],[35,15781,14973],{"className":15782,"style":14999},[78,79],[35,15784,15003],{"className":15785},[78,79],[35,15787,238],{"className":15788},[278],[35,15790,56],{"className":15791},[78,79],[35,15793,248],{"className":15794},[296],[35,15796,128],{"className":15797},[78],[35,15799],{"className":15800,"style":166},[165],[35,15802,136],{"className":15803},[170],[35,15805],{"className":15806,"style":166},[165],[35,15808,15810,15813],{"className":15809},[69],[35,15811],{"className":15812,"style":312},[73],[35,15814,573],{"className":15815},[78],[471,15817,15818,15906,15912,15998,16073,16104],{},[464,15819,15543,15820,722],{},[35,15821,15823,15855],{"className":15822},[38],[35,15824,15826],{"className":15825},[42],[44,15827,15828],{"xmlns":46},[48,15829,15830,15852],{},[51,15831,15832,15834,15836,15838,15840,15842,15844,15846,15848,15850],{},[54,15833,128],{"mathvariant":127},[54,15835,56],{},[54,15837,15104],{"mathvariant":127},[54,15839,6464],{},[134,15841,238],{"stretchy":237},[54,15843,56],{},[134,15845,248],{"stretchy":237},[54,15847,128],{"mathvariant":127},[134,15849,136],{},[252,15851,573],{},[58,15853,15854],{"encoding":60},"|G\u002FZ(G)| = 2",[35,15856,15858,15897],{"className":15857,"ariaHidden":65},[64],[35,15859,15861,15864,15867,15870,15873,15876,15879,15882,15885,15888,15891,15894],{"className":15860},[69],[35,15862],{"className":15863,"style":152},[73],[35,15865,128],{"className":15866},[78],[35,15868,56],{"className":15869},[78,79],[35,15871,15104],{"className":15872},[78],[35,15874,6464],{"className":15875,"style":8683},[78,79],[35,15877,238],{"className":15878},[278],[35,15880,56],{"className":15881},[78,79],[35,15883,248],{"className":15884},[296],[35,15886,128],{"className":15887},[78],[35,15889],{"className":15890,"style":166},[165],[35,15892,136],{"className":15893},[170],[35,15895],{"className":15896,"style":166},[165],[35,15898,15900,15903],{"className":15899},[69],[35,15901],{"className":15902,"style":312},[73],[35,15904,573],{"className":15905},[78],[464,15907,15908,15909,722],{},"这里有一个著名的结论：",[30,15910,15911],{},"“循环群判定法”",[464,15913,15914,15915,15968,15969,15997],{},"如果商群 ",[35,15916,15918,15941],{"className":15917},[38],[35,15919,15921],{"className":15920},[42],[44,15922,15923],{"xmlns":46},[48,15924,15925,15939],{},[51,15926,15927,15929,15931,15933,15935,15937],{},[54,15928,56],{},[54,15930,15104],{"mathvariant":127},[54,15932,6464],{},[134,15934,238],{"stretchy":237},[54,15936,56],{},[134,15938,248],{"stretchy":237},[58,15940,15570],{"encoding":60},[35,15942,15944],{"className":15943,"ariaHidden":65},[64],[35,15945,15947,15950,15953,15956,15959,15962,15965],{"className":15946},[69],[35,15948],{"className":15949,"style":152},[73],[35,15951,56],{"className":15952},[78,79],[35,15954,15104],{"className":15955},[78],[35,15957,6464],{"className":15958,"style":8683},[78,79],[35,15960,238],{"className":15961},[278],[35,15963,56],{"className":15964},[78,79],[35,15966,248],{"className":15967},[296]," 是循环群，那么 ",[35,15970,15972,15985],{"className":15971},[38],[35,15973,15975],{"className":15974},[42],[44,15976,15977],{"xmlns":46},[48,15978,15979,15983],{},[51,15980,15981],{},[54,15982,56],{},[58,15984,56],{"encoding":60},[35,15986,15988],{"className":15987,"ariaHidden":65},[64],[35,15989,15991,15994],{"className":15990},[69],[35,15992],{"className":15993,"style":74},[73],[35,15995,56],{"className":15996},[78,79]," 一定是交换群。",[464,15999,16000,16001,16072],{},"阶数为 2 的群一定是循环群（同构于 ",[35,16002,16004,16022],{"className":16003},[38],[35,16005,16007],{"className":16006},[42],[44,16008,16009],{"xmlns":46},[48,16010,16011,16019],{},[51,16012,16013],{},[2203,16014,16015,16017],{},[54,16016,6464],{"mathvariant":3928},[252,16018,573],{},[58,16020,16021],{"encoding":60},"\\mathbb{Z}_2",[35,16023,16025],{"className":16024,"ariaHidden":65},[64],[35,16026,16028,16032],{"className":16027},[69],[35,16029],{"className":16030,"style":16031},[73],"height:0.8389em;vertical-align:-0.15em;",[35,16033,16035,16038],{"className":16034},[78],[35,16036,6464],{"className":16037},[78,3954],[35,16039,16041],{"className":16040},[363],[35,16042,16044,16064],{"className":16043},[367,2234],[35,16045,16047,16061],{"className":16046},[371],[35,16048,16050],{"className":16049,"style":2241},[375],[35,16051,16052,16055],{"style":2244},[35,16053],{"className":16054,"style":383},[382],[35,16056,16058],{"className":16057},[387,388,389,390],[35,16059,573],{"className":16060},[78,390],[35,16062,2258],{"className":16063},[2257],[35,16065,16067],{"className":16066},[371],[35,16068,16070],{"className":16069,"style":2265},[375],[35,16071],{},"）。",[464,16074,16075,16076,14689],{},"所以 ",[35,16077,16079,16092],{"className":16078},[38],[35,16080,16082],{"className":16081},[42],[44,16083,16084],{"xmlns":46},[48,16085,16086,16090],{},[51,16087,16088],{},[54,16089,56],{},[58,16091,56],{"encoding":60},[35,16093,16095],{"className":16094,"ariaHidden":65},[64],[35,16096,16098,16101],{"className":16097},[69],[35,16099],{"className":16100,"style":74},[73],[35,16102,56],{"className":16103},[78,79],[464,16105,16106,16110,16111,16139,16140,16206,16207,16260,16261,14689],{},[16107,16108,16109],"em",{},"进一步补充","：其实如果 ",[35,16112,16114,16127],{"className":16113},[38],[35,16115,16117],{"className":16116},[42],[44,16118,16119],{"xmlns":46},[48,16120,16121,16125],{},[51,16122,16123],{},[54,16124,56],{},[58,16126,56],{"encoding":60},[35,16128,16130],{"className":16129,"ariaHidden":65},[64],[35,16131,16133,16136],{"className":16132},[69],[35,16134],{"className":16135,"style":74},[73],[35,16137,56],{"className":16138},[78,79]," 是交换群，",[35,16141,16143,16167],{"className":16142},[38],[35,16144,16146],{"className":16145},[42],[44,16147,16148],{"xmlns":46},[48,16149,16150,16164],{},[51,16151,16152,16154,16156,16158,16160,16162],{},[54,16153,6464],{},[134,16155,238],{"stretchy":237},[54,16157,56],{},[134,16159,248],{"stretchy":237},[134,16161,136],{},[54,16163,56],{},[58,16165,16166],{"encoding":60},"Z(G)=G",[35,16168,16170,16197],{"className":16169,"ariaHidden":65},[64],[35,16171,16173,16176,16179,16182,16185,16188,16191,16194],{"className":16172},[69],[35,16174],{"className":16175,"style":152},[73],[35,16177,6464],{"className":16178,"style":8683},[78,79],[35,16180,238],{"className":16181},[278],[35,16183,56],{"className":16184},[78,79],[35,16186,248],{"className":16187},[296],[35,16189],{"className":16190,"style":166},[165],[35,16192,136],{"className":16193},[170],[35,16195],{"className":16196,"style":166},[165],[35,16198,16200,16203],{"className":16199},[69],[35,16201],{"className":16202,"style":74},[73],[35,16204,56],{"className":16205},[78,79],"，那么 ",[35,16208,16210,16233],{"className":16209},[38],[35,16211,16213],{"className":16212},[42],[44,16214,16215],{"xmlns":46},[48,16216,16217,16231],{},[51,16218,16219,16221,16223,16225,16227,16229],{},[54,16220,56],{},[54,16222,15104],{"mathvariant":127},[54,16224,6464],{},[134,16226,238],{"stretchy":237},[54,16228,56],{},[134,16230,248],{"stretchy":237},[58,16232,15570],{"encoding":60},[35,16234,16236],{"className":16235,"ariaHidden":65},[64],[35,16237,16239,16242,16245,16248,16251,16254,16257],{"className":16238},[69],[35,16240],{"className":16241,"style":152},[73],[35,16243,56],{"className":16244},[78,79],[35,16246,15104],{"className":16247},[78],[35,16249,6464],{"className":16250,"style":8683},[78,79],[35,16252,238],{"className":16253},[278],[35,16255,56],{"className":16256},[78,79],[35,16258,248],{"className":16259},[296]," 只有 1 个元素。所以\"情况2\"实际上是不可能发生的，但无论如何，推导都指向 ",[35,16262,16264,16277],{"className":16263},[38],[35,16265,16267],{"className":16266},[42],[44,16268,16269],{"xmlns":46},[48,16270,16271,16275],{},[51,16272,16273],{},[54,16274,56],{},[58,16276,56],{"encoding":60},[35,16278,16280],{"className":16279,"ariaHidden":65},[64],[35,16281,16283,16286],{"className":16282},[69],[35,16284],{"className":16285,"style":74},[73],[35,16287,56],{"className":16288},[78,79],{"title":16290,"searchDepth":16291,"depth":16291,"links":16292},"",2,[16293,16295,16297],{"id":19,"depth":16294,"text":20},3,{"id":6544,"depth":16294,"text":16296},"第二部分：置换群 SnS_nSn​ 与 AnA_nAn​（计算篇）",{"id":8519,"depth":16294,"text":8520},"md",{"order":16300},1,true,"\u002Fsource\u002Fmath\u002Fabstract_algebra\u002Fgroupproblem",{"title":5,"description":16290},"source\u002Fmath\u002Fabstract_algebra\u002Fgroupproblem","3RDZDVbG7TptQ_QHahOZdABszVwE9KCsjpjqOzR-WEI",null,1784032893732]