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In [[mathematics]], a [[group (mathematics)|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: 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 simple group ''S'' such that <math>S \leq A \leq \operatorname{Aut}(S).</math> | |||
== Examples == | |||
* Trivially, nonabelian simple groups and the full group of automorphisms are almost simple, but proper examples exist, meaning almost simple groups that are neither simple nor the full automorphism group. | |||
* For <math>n\geq 5,</math> the [[symmetric group]] <math>S_n</math> is almost simple, with the simple group being the [[alternating group]] <math>A_n.</math> For <math>n \neq 6,</math> <math>S_n</math> is the full automorphism group of <math>A_n,</math> while for <math>n = 6,</math> <math>S_6</math> sits properly between <math>A_6</math> and <math>\operatorname{Aut}(A_6),</math> due to the [[Automorphisms of the symmetric and alternating groups#exceptional outer automorphism|exceptional outer automorphism]] of <math>A_6.</math> | |||
== Properties == | |||
The full automorphism group of a nonabelian simple group is a [[complete group]] (the conjugation map is an isomorphism to the automorphism 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 [[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]] | |||
Revision as of 10:22, 13 November 2013
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: 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 simple group S such that
Examples
- Trivially, nonabelian simple groups and the full group of automorphisms are almost simple, but proper examples exist, meaning almost simple groups that are neither simple nor the full automorphism group.
- For the symmetric group is almost simple, with the simple group being the alternating group For is the full automorphism group of while for sits properly between and due to the exceptional outer automorphism of
Properties
The full automorphism group of a nonabelian simple group is a complete group (the conjugation map is an isomorphism to the automorphism 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 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
Notes
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External links
- Almost simple group at the Group Properties wiki