BREAKING NEWS

## Summary

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 $S\leq A\leq \operatorname {Aut} (S).$ ## 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 $n=5$  or $n\geq 7,$  the symmetric group $\mathrm {S} _{n}$  is the automorphism group of the simple alternating group $\mathrm {A} _{n},$  so $\mathrm {S} _{n}$  is almost simple in this trivial sense.
• For $n=6$  there is a proper example, as $\mathrm {S} _{6}$  sits properly between the simple $\mathrm {A} _{6}$  and $\operatorname {Aut} (\mathrm {A} _{6}),$  due to the exceptional outer automorphism of $\mathrm {A} _{6}.$  Two other groups, the Mathieu group $\mathrm {M} _{10}$  and the projective general linear group $\operatorname {PGL} _{2}(9)$  also sit properly between $\mathrm {A} _{6}$  and $\operatorname {Aut} (\mathrm {A} _{6}).$

## 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.