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In mathematics, the outer automorphism group of a group ''G'' is the quotient Aut(''G'') / Inn(''G''), where Aut(''G'') is the automorphism group of ''G'' and Inn(''G'') is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted Out(''G''). If Out(''G'') is trivial and ''G'' has a trivial center, then ''G'' is said to be complete. An automorphism of a group which is not inner is called an outer automorphism. Note that the elements of Out(''G'') are cosets of automorphisms of ''G'', and not themselves automorphisms; this is an instance of the fact that quotients of groups are not in general (isomorphic to) subgroups. Elements of Out(''G'') are cosets of Inn(''G'') in Aut(''G''). For example, for the alternating group ''A''''n'', the outer automorphism group is usually the group of order 2, with exceptions noted below. Considering ''A''''n'' as a subgroup of the symmetric group ''S''''n'' conjugation by any odd permutation is an outer automorphism of ''A''''n'' or more precisely "represents the class of the (non-trivial) outer automorphism of ''A''''n''", but the outer automorphism does not correspond to conjugation by any ''particular'' odd element, and all conjugations by odd elements are equivalent up to conjugation by an even element. However, for an abelian group ''A,'' the inner automorphism group is trivial and thus the automorphism group and outer automorphism group are naturally identified, and outer automorphisms do act on ''A''. ==Out(''G'') for some finite groups== For the outer automorphism groups of all finite simple groups see the list of finite simple groups. Sporadic simple groups and alternating groups (other than the alternating group ''A''6; see below) all have outer automorphism groups of order 1 or 2. The outer automorphism group of a finite simple group of Lie type is an extension of a group of "diagonal automorphisms" (cyclic except for D''n''(''q'') when it has order 4), a group of "field automorphisms" (always cyclic), and a group of "graph automorphisms" (of order 1 or 2 except for D4(''q'') when it is the symmetric group on 3 points). These extensions are not always semidirect products, as the case of the alternating group A6 shows; a precise criterion for this to happen is given in : A. Lucchini, F. Menegazzo and M. Morigi, On the existence of a complement for a finite simple group in its automorphism group, Illinois J. Math. 47 (2003), 395-418. \left(1-\frac\right) elements; one corresponding to multiplication by an invertible element in Z''n'' viewed as a ring. |- | Z''p''''n'' | ''p'' prime, ''n'' > 1 | GL''n''(''p'') |(''p''''n'' − 1)(''p''''n'' − ''p'' )(''p''''n'' − ''p''2) ... (''p''''n'' − ''p''''n''−1) elements |- | ''S''''n'' | n ≠ 6 || trivial | 1 |- | ''S''6 | || Z2 (see below) | 2 |- | ''A''''n'' | ''n'' ≠ 6 || Z2 | 2 |- | ''A''6 | | Z2 × Z2(see below) | 4 |- | PSL2(''p'') | ''p'' > 3 prime || Z2 |2 |- | PSL2(2''n'') | ''n'' > 1 || Z''n'' |''n'' |- | PSL3(4) = M21 | || Dih6 | 12 |- | M''n'' | ''n'' = 11, 23, 24 || trivial |1 |- | M''n'' | ''n'' = 12, 22 || Z2 |2 |- | Co''n'' | ''n'' = 1, 2, 3 || trivial |1 |} 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「Outer automorphism group」の詳細全文を読む スポンサード リンク
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