# 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: 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 simple group S such that $S \leq A \leq \operatorname{Aut}(S).$

## Examples

• Trivially, nonabelian 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 $n\geq 5,$ the symmetric group $S_n$ is almost simple, with the simple group being the alternating group $A_n.$ For $n \neq 6,$ $S_n$ is the full automorphism group of $A_n,$ while for $n = 6,$ $S_6$ sits properly between $A_6$ and $\operatorname{Aut}(A_6),$ due to the exceptional outer automorphism of $A_6.$

## Properties

The full automorphism group of a nonabelian 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.