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交代群

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H 2 ( A n , Z ) = 0 for n = 1 , 2 , 3 ; {\displaystyle H_{2}(A_{n},\mathbb {Z} )=0{\mbox{ for }}n=1,2,3;} H 2 ( A n , Z ) = Z / 2 Z for n = 4 , 5 ; {\displaystyle H_{2}(A_{n},\mathbb {Z} )=\mathbb {Z} /2\mathbb {Z} {\mbox{ for }}n=4,5;} H 2 ( A n , Z ) = Z / 6 Z for n = 6 , 7 ; {\displaystyle H_{2}(A_{n},\mathbb {Z} )=\mathbb {Z} /6\mathbb {Z} {\mbox{ for }}n=6,7;} H 2 ( A n , Z ) = Z / 2 Z for n ≥ 8. {\displaystyle H_{2}(A_{n},\mathbb {Z} )=\mathbb {Z} /2\mathbb {Z} {\mbox{ for }}n\geq 8.}
脚注 ^ Scott 1987, p. 267.
^ たとえば基本交代式（差積） Δ ( x 1 , … , x n ) := ∏ 1 ≤ i < j ≤ n ( x i − x j ) {\displaystyle \Delta (x_{1},\dotsc ,x_{n}):=\prod _{1\leq i<j\leq n}(x_{i}-x_{j})}
^ Vilyams 2001.
^ ガロワは素数位数でない単純群の最小位数は 60 であると予想していた (Kline 1992, p. 766)。
^ Scott 1987, pp. 298?300, §11.1 Conjugacy classes.
^ Wilson 2009, p. 18（Theorem 2.3, Exercise 2.16 参照）
^ a b c Scott 1987, p. 295.
^ a b Coxeter, Harold S. M.; Moser, William O. J. (1980). Generators and Relations for Discrete Groups (4th ed.). p. 66. .mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation.cs-ja1 q,.mw-parser-output .citation.cs-ja2 q{quotes:"「""」""『""』"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free a,.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription a,.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:#d33}.mw-parser-output .cs1-visible-error{color:#d33}.mw-parser-output .cs1-maint{display:none;color:#3a3;margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}ISBN 978-3-662-21945-4
^ Huppert 1967, p. 138, Beispiel 19.8（Aufgaben 75, Beispiel 19.9 も参照）
^ Wilson, Robert (October 31, 2006), ⇒“Chapter 2: Alternating groups”, ⇒http://www.maths.qmul.ac.uk/~raw/fsgs.html, 2.7: Covering groups

参考文献.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-100{font-size:100%}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}

Huppert, B. (1967), Endliche Gruppen I, Die Grundlehren der Mathematischen Wissenschaften, 134, Springer-Verlag, doi:10.1007/978-3-642-64981-3, ISBN 978-3-642-64982-0, MR0224703, Zbl 0217.07201, https://books.google.co.jp/books?id=dfunBgAAQBAJ

Kline, Morris (1972), Mathematical Thought from Ancient to Modern Times, Oxford University Press, ISBN 0-19-501496-0, MR0472307, Zbl 0277.01001 (Review by Gian-Carlo Rota in Bull. Amer. Math. Soc.)

Schur, Issai (1911), “Uber die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen”, J. Reine Angew. Math. 139: 155?250, doi:10.1515/crll.1911.139.155

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