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

Examples

Trivially, non-abelian simple groups and the full group of automorphisms are almost simple. For $n=5$ or $n \geq 7,$ the symmetric group $\mathrm{S}_n$ is the automorphism group of the simple alternating group $\mathrm{A}_n,$ so $\mathrm{S}_n$ is almost simple in this trivial sense.
For $n=6$ there is a proper example, as $\mathrm{S}_6$ sits properly between the simple $\mathrm{A}_6$ and $\operatorname{Aut}(\mathrm{A}_6),$ due to the exceptional outer automorphism of $\mathrm{A}_6.$ Two other groups, the Mathieu group $\mathrm{M}_{10}$ and the projective general linear group $\operatorname{PGL}_2(9)$ also sit properly between $\mathrm{A}_6$ and $\operatorname{Aut}(\mathrm{A}_6).$

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),{{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 }} 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.

Notes

External links

[http://groupprops.subwiki.org/wiki/Almost_simple_group Almost simple group] at the Group Properties wiki

Category:Properties of groups