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 (non-abelian) simple group and its automorphism group. In symbols, a group ${\displaystyle A}$ is almost simple if there is a (non-abelian) simple group S such that ${\displaystyle S\leq A\leq \operatorname {Aut} (S)}$, where the inclusion of ${\displaystyle S}$ in ${\displaystyle \mathrm {Aut} (S)}$ is the action by conjugation, which is faithful since ${\displaystyle S}$ has a trivial center.¹

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

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.