# Almost simple group

In mathematics, a group is said to be almost simple if it contains a non-abelian simple group and is contained within the automorphism group of that simple group – that is, if it fits between a (non-abelian) simple group and its automorphism group. In symbols, a group A is almost simple if there is a (non-abelian) simple group S such that $\displaystyle{ S \leq A \leq \operatorname{Aut}(S). }$

## Examples

• Trivially, non-abelian simple groups and the full group of automorphisms are almost simple, but proper examples exist, meaning almost simple groups that are neither simple nor the full automorphism group.
• For $\displaystyle{ n=5 }$ or $\displaystyle{ n \geq 7, }$ the symmetric group $\displaystyle{ \mathrm{S}_n }$ is the automorphism group of the simple alternating group $\displaystyle{ \mathrm{A}_n, }$ so $\displaystyle{ \mathrm{S}_n }$ is almost simple in this trivial sense.
• For $\displaystyle{ n=6 }$ there is a proper example, as $\displaystyle{ \mathrm{S}_6 }$ sits properly between the simple $\displaystyle{ \mathrm{A}_6 }$ and $\displaystyle{ \operatorname{Aut}(\mathrm{A}_6), }$ due to the exceptional outer automorphism of $\displaystyle{ \mathrm{A}_6. }$ Two other groups, the Mathieu group $\displaystyle{ \mathrm{M}_{10} }$ and the projective general linear group $\displaystyle{ \operatorname{PGL}_2(9) }$ also sit properly between $\displaystyle{ \mathrm{A}_6 }$ and $\displaystyle{ \operatorname{Aut}(\mathrm{A}_6). }$

## Properties

The full automorphism group of a non-abelian simple group is a complete group (the conjugation map is an isomorphism to the automorphism group), but proper subgroups of the full automorphism group need not be complete.

## Structure

By the Schreier conjecture, now generally accepted as a corollary of the classification of finite simple groups, the outer automorphism group of a finite simple group is a solvable group. Thus a finite almost simple group is an extension of a solvable group by a simple group.