Geometric invariant theory: Difference between revisions

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In [[mathematics]], a '''quasisimple group''' (also known as a '''covering group''') is a [[group (mathematics)|group]] that is a [[perfect group|perfect]] [[Central extension (mathematics)|central extension]] ''E'' of a [[simple group]] ''S''.  In other words, there is a [[short exact sequence]]
 
:1 &rarr; ''Z''(''E'') &rarr; ''E'' &rarr; ''S'' &rarr; 1
 
such that ''E'' = [''E'', ''E''], where ''Z''(''E'') denotes the [[Center (group theory)|center]] of ''E'' and [, ] denotes the [[commutator]].<ref>I. Martin Isaacs, ''Finite group theory'' (2008), p. 272.</ref>
 
Equivalently, a group is quasisimple if it is isomorphic to its [[commutator subgroup]] and its [[inner automorphism|inner automorphism group]] Inn(''G'') (its [[quotient]] by its center) is simple; due to [[Perfect_group#Grün's lemma|Grün's lemma]], Inn(''G'') must be [[non-abelian group|non-abelian]]. All non-abelian simple groups are quasisimple.
 
The [[subnormal subgroup|subnormal]] quasisimple subgroups of a group control the structure of a finite [[insoluble group]] in much the same way as the minimal [[normal subgroup]]s of a finite [[soluble group]] do, and so are given a name, [[component (group theory)|component]]. 
 
The subgroup generated by the subnormal quasisimple subgroups is called the '''layer''', and along with the minimal normal soluble subgroups generates a subgroup called the [[generalized Fitting subgroup]].
 
The quasisimple groups are often studied alongside the simple groups and groups related to their [[automorphism group]]s, the [[almost simple group]]s. The [[representation theory of finite groups|representation theory]] of the quasisimple groups is nearly identical to the [[projective representation|projective representation theory]] of the simple groups.
 
==Examples==
The [[covering groups of the alternating and symmetric groups|covering groups of the alternating groups]] are quasisimple but not simple, for <math>n \geq 5.</math>
 
==See also==
* [[Almost simple group]]
* [[Schur multiplier]]
* [[Semisimple group]]
 
==References==
* {{ cite book | last=Aschbacher | first=Michael | title=Finite Group Theory | publisher=[[Cambridge University Press]] | year=2000 | isbn=0-521-78675-4 | zbl=0997.20001 }}
 
==External links==
*http://mathworld.wolfram.com/QuasisimpleGroup.html
 
==Notes==
{{Reflist}}
 
[[Category:Group theory]]
[[Category:Properties of groups]]
 
 
{{Abstract-algebra-stub}}

Latest revision as of 14:42, 10 January 2015



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