Föppl–von Kármán equations: Difference between revisions

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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>
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== 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]]

Latest revision as of 10:59, 13 June 2014

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