Complete group
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In mathematics, a group G is said to be complete if all the automorphisms of G are inner, and the group is a centerless group, that is, it has a trivial center. Thus, there is a natural isomorphism between the group and its automorphism group, with each group element giving the automorphism obtained via conjugation by it.
As an example, all the symmetric groups Sn are complete except when n = 2 or 6. For the case n = 2 the group has a nontrivial center, while for the case n = 6 there is an outer automorphism.
For a simple group G, the automorphism group G is complete, i.e. Inn(Aut(G)) = Aut(Aut(G)). The automorphism group of a simple group is termed an almost simple group.
More examples would be useful... Template:Math-stub