# 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 ${\displaystyle S\leq A\leq \operatorname {Aut} (S).}$

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

## 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.