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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク List of finite simple groups ： ウィキペディア英語版 List of finite simple groups In mathematics, the classification of finite simple groups states that every finite simple group is cyclic, or alternating, or in one of 16 families of groups of Lie type, or one of 26 sporadic groups. The list below gives all finite simple groups, together with their order, the size of the Schur multiplier, the size of the outer automorphism group, usually some small representations, and lists of all duplicates. == Summary == The following table is a complete list of the 18 families of finite simple groups and the 26 sporadic simple groups, along with their orders. Any non-simple members of each family are listed, as well as any members duplicated within a family or between families. (In removing duplicates it is useful to note that no two finite simple groups have the same order, except that the group A8 = ''A''3(2) and ''A''2(4) both have order 20160, and that the group ''Bn''(''q'') has the same order as ''Cn''(''q'') for ''q'' odd, ''n'' > 2. The smallest of the latter pairs of groups are ''B''3(3) and ''C''3(3) which both have order 4585351680.) There is an unfortunate conflict between the notations for the alternating groups A''n'' and the groups of Lie type ''An''(''q''). Some authors use various different fonts for A''n'' to distinguish them. In particular, in this article we make the distinction by setting the alternating groups A''n'' in Roman font and the Lie-type groups ''An''(''q'') in italic. In what follows, ''n'' is a positive integer, and ''q'' is a positive power of a prime number ''p'', with the restrictions noted. | | A5≈''A''1(4)≈''A''1(5) A6≈''A''1(9) A8≈''A''3(2) |- |colspan=6| |- |rowspan=4| Classical Chevalley groups | ''A''''n''(''q'') | $\backslash frac\; n(n+1)\}\}\; \backslash prod\_^n(q^-1)$ | ''A''1(2), ''A''1(3) | ''A''1(4)≈''A''1(5)≈A5 ''A''1(7)≈''A''2(2) ''A''1(9)≈A6 ''A''3(2)≈A8 |- | ''B''''n''(''q'') ''n'' > 1 ||$\backslash frac\backslash prod\_^n(q^-1)$ | ''B''2(2) | ''Bn''(2''m'')≈''Cn''(2''m'') ''B''2(3)≈2''A''3(22) |- | ''C''''n''(''q'') ''n'' > 2 ||$\backslash frac\backslash prod\_^n(q^-1)$ | | ''Cn''(2''m'')≈''Bn''(2''m'') |- | ''D''''n''(''q'') ''n'' > 3 | $\backslash frac\backslash prod\_^(q^-1)$ | | |- |rowspan=5| Exceptional Chevalley groups | ''E''6(''q'') | $\backslash frac\backslash prod\_(q^i-1)$ | | |- | ''E''7(''q'') | $\backslash frac\backslash prod\_(q^i-1)$ | | |- | ''E''8(''q'') | $q^\; \backslash prod\_(q^i-1)$ | | |- | ''F''4(''q'') | $q^\; \backslash prod\_(q^i-1)$ | | |- | ''G''2(''q'') | $q^6\; \backslash prod\_(q^i-1)$ | ''G''2(2) | |- |rowspan=2| Classical Steinberg groups | 2''A''''n''(''q''2) ''n'' > 1 | $\backslash frac\backslash prod\_^n(q^-(-1)^)$ | 2''A''2(22) | 2''A''3(22)≈''B''2(3) |- ||2''Dn''(''q''2) ''n'' > 3 | $\backslash frac(q^n+1)\backslash prod\_^(q^-1)$ | | |- |rowspan=2| Exceptional Steinberg groups ||2''E6''(''q''2) | $\backslash frac\backslash prod\_(q^i-(-1)^i)$ | | |- ||3''D4''(''q''3) | $q^(q^8+q^4+1)(q^6-1)(q^2-1)$ | | |- ||Suzuki groups | 2''B''2(''q'') ''q'' = 22''n''+1 n≥ 1 | $q^2(q^2+1)(q-1)$ | | |- |rowspan=3| Ree groups + Tits group | 2''F''4(''q'') ''q'' = 22''n''+1 n≥ 1 | $q^(q^6+1)(q^4-1)(q^3+1)(q-1)$ | | |- | 2''F''4(2)' |colspan=3| 212(26 + 1)(24 − 1)(23 + 1)(2 − 1)/2 = |- | 2''G''2(''q'') ''q'' = 32''n''+1 n≥ 1 | $q^3(q^3+1)(q-1)$ | | |- |colspan=6| |- |rowspan=5|Mathieu groups | M11 |colspan=3| |- | M12 |colspan=3| |- | M22 |colspan=3| |- | M23 |colspan=3| |- | M24 |colspan=3| |- |rowspan=4|Janko groups | J1 |colspan=3| |- | J2 |colspan=3| |- | J3 |colspan=3| |- | J4 |colspan=3| |- |rowspan=3|Conway groups | Co3 |colspan=3| |- | Co2 |colspan=3| |- | Co1 |colspan=3| |- |rowspan=3|Fischer groups | Fi22 |colspan=3| |- | Fi23 |colspan=3| |- | Fi24' |colspan=3| |- | Higman–Sims group | HS |colspan=3| |- | McLaughlin group | McL |colspan=3| |- | Held group | He |colspan=3| |- | Rudvalis group | Ru |colspan=3| |- | Suzuki sporadic group | Suz |colspan=3| |- | O'Nan group | O'N |colspan=3| |- | Harada–Norton group | HN |colspan=3| |- | Lyons group | Ly |colspan=3| |- | Thompson group | Th |colspan=3| |- | Baby Monster group | B |colspan=3| |- | Monster group | M |colspan=3| |} 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「List of finite simple groups」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.