Examples
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.
The automorphism group of a simple group G is an almost simple group; for a nonabelian simple group G, the automorphism group of G is complete.
Read more about this topic: Complete Group
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