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 <math>A</math> is almost simple if there is a (non-abelian) simple group ''S'' such that <math>S \leq A \leq \operatorname{Aut}(S)</math>, where the inclusion of <math>S</math> in <math>\mathrm{Aut}(S)</math> is the action by conjugation, which is faithful since <math>S</math> has a trivial center.<ref>{{Cite journal |last1=Dallavolta |first1=F. |last2=Lucchini |first2=A. |date=1995-11-15 |title=Generation of Almost Simple Groups |url=https://www.sciencedirect.com/science/article/pii/S0021869385713452 |journal=Journal of Algebra |volume=178 |issue=1 |pages=194–223 |doi=10.1006/jabr.1995.1345 |issn=0021-8693|url-access=subscription }}</ref>
== Examples == * Trivially, non-abelian simple groups and the full group of automorphisms are almost simple. For <math>n=5</math> or <math>n \geq 7,</math> the symmetric group <math>\mathrm{S}_n</math> is the automorphism group of the simple alternating group <math>\mathrm{A}_n,</math> so <math>\mathrm{S}_n</math> is almost simple in this trivial sense. * For <math>n=6</math> there is a proper example, as <math>\mathrm{S}_6</math> sits properly between the simple <math>\mathrm{A}_6</math> and <math>\operatorname{Aut}(\mathrm{A}_6),</math> due to the exceptional outer automorphism of <math>\mathrm{A}_6.</math> Two other groups, the Mathieu group <math>\mathrm{M}_{10}</math> and the projective general linear group <math>\operatorname{PGL}_2(9)</math> also sit properly between <math>\mathrm{A}_6</math> and <math>\operatorname{Aut}(\mathrm{A}_6).</math>
== 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),<ref>{{Citation |last=Robinson |first=Derek J. S. |title=Subnormal Subgroups |date=1996 |work=A Course in the Theory of Groups |series=Graduate Texts in Mathematics |volume=80 |editor-last=Robinson |editor-first=Derek J. S. |url=https://link.springer.com/chapter/10.1007/978-1-4419-8594-1_13 |access-date=2024-11-23 |at=Corollary 13.5.10 |place=New York, NY |publisher=Springer |language=en |doi=10.1007/978-1-4419-8594-1_13 |isbn=978-1-4419-8594-1|url-access=subscription }}</ref> 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.
== See also == * Quasisimple group * Semisimple group
== Notes == {{reflist|group=note}}
== External links == * [http://groupprops.subwiki.org/wiki/Almost_simple_group Almost simple group] at the Group Properties wiki
Category:Properties of groups