Absolutely simple group: Difference between revisions

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In [[mathematics]], in the field of [[group theory]], a [[subgroup]] of a [[group (mathematics)|group]] is termed a [[retract]] if there is an [[endomorphism]] of the group that maps [[surjective]]ly to the subgroup and is identity on the subgroup. In symbols, <math>H</math> is a retract of <math>G</math> if and only if there is an endomorphism <math>\sigma:G \to G</math> such that <math>\sigma(h) = h</math> for all <math>h \in H</math> and <math>\sigma(g) \in H</math> for all <math>g \in G</math>.
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The endomorphism itself is termed an [[idempotent]] endomorphism or a retraction.
 
The following is known about retracts:
 
* A subgroup is a retract if and only if it has a [[normal subgroup|normal]] [[complement (group theory)|complement]]. The normal complement, specifically, is the kernel of the retraction.
* Every [[direct product of groups|direct factor]] is a retract. Conversely, any retract which is a normal subgroup is a direct factor.
* Every retract has the [[CEP subgroup|congruence extension property]].
* Every [[regular factor]], and in particular, every [[free factor]], is a retract.
 
==References==
{{unreferenced|date=September 2008}}
 
[[Category:Group theory]]
[[Category:Subgroup properties]]
 
 
{{Abstract-algebra-stub}}

Latest revision as of 14:53, 13 November 2014

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