# Almost simple not implies simple or complete

## Contents

## 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 that is not complete, and also such that is *not* a maximal normal subgroup of . In other words, there exist intermediate normal subgroups between and .

## Proof

### Example of the alternating group of degree six

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

Let be the alternating group of degree six and be the symmetric group of degree six. The automorphism group of is a group containing with index two, namely, a semidirect product of by an outer automorphism of order two. Thus, is not a maximal subgroup of -- is an intermediate subgroup.

### Example of projective special linear groups

If is a power of an odd prime such that , then the group has index four inside , and there is an intermediate subgroup. Since is itself contained in the automorphism group of the simple group , the intermediate subgroup is a subgroup that is neither equal to the simple group nor equal to the whole automorphism group.