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Complete group
In mathematics, a group ''G'' is said to be complete if every automorphism of ''G'' is inner, and it is centerless; that is, it has a trivial outer automorphism group and trivial center.
Equivalently, a group is complete if the conjugation map, ''G'' → Aut(''G'') (sending an element ''g'' to conjugation by ''g''), is an isomorphism: injectivity implies the group is centerless, as no inner automorphisms are the identity, while surjectivity implies it has no outer automorphisms.
== Examples ==
As an example, all the symmetric groups S''n'' are complete except when ''n'' = 2 or 6. For the case ''n'' = 2 the group has a non-trivial center, while for the case ''n'' = 6 there is an outer automorphism.
The automorphism group of a simple group ''G'' is an almost simple group; for a non-abelian simple group ''G'', the automorphism group of ''G'' is complete.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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