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.
is the action by conjugation, which is faithful since
[1] The full automorphism group of a non-abelian simple group is a complete group (the conjugation map is an isomorphism to the automorphism group),[2] but proper subgroups of the full automorphism group need not be complete.
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.