Almost simple not implies simple or complete

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Statement

There can exist a finite almost simple group that is neither a simple group nor a complete group.

In other words, there can exist a finite simple non-Abelian group S that is not complete, and also such that S is not a maximal normal subgroup of \operatorname{Aut}(S). In other words, there exist intermediate normal subgroups between S and \operatorname{Aut}(S).

Proof

Example of the alternating group of degree six

Further information: Alternating group:A6, symmetric group:S6

Let S be the alternating group of degree six and G be the symmetric group of degree six. The automorphism group of S is a group containing G with index two, namely, a semidirect product of G by an outer automorphism of order two. Thus, S is not a maximal subgroup of \operatorname{Aut}(S) -- G is an intermediate subgroup.

Example of projective special linear groups

If q is a power of an odd prime such that 4 | q -1, then the group PSL(4,q) has index four inside PGL(4,q), and there is an intermediate subgroup. Since PGL(4,q) is itself contained in the automorphism group of the simple group PSL(4,q), the intermediate subgroup is a subgroup that is neither equal to the simple group nor equal to the whole automorphism group.