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 simple group and its automorphism group. In symbols, a group is almost simple if there is a simple group S such that, where the inclusion of in is the action by conjugation, which is faithful since has a trivial center.

Examples

Trivially, non-abelian simple groups and the full group of automorphisms are almost simple. For or the symmetric group is the automorphism group of the simple alternating group so is almost simple in this trivial sense.
For there is a proper example, as sits properly between the simple and due to the Automorphisms of the [symmetric and alternating groups#exceptional outer automorphism|exceptional outer automorphism] of Two other groups, the Mathieu group and the projective general linear group also sit properly between and

Properties

The full automorphism group of a non-abelian simple group is a complete 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 finite simple groups">finite group">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.