Simple subnormal subgroup - Revision history

Vipul at 20:12, 12 June 2008

2008-06-12T20:12:25Z

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	* It is ''minimal'' among subnormal subgroups, viz there is no smaller nontrivial subgroup of it that is subnormal in the whole group		* It is ''minimal'' among subnormal subgroups, viz there is no smaller nontrivial subgroup of it that is subnormal in the whole group

	{{obtainedbyapplyingthe\|minimal operator\|~~subnormality~~}}		{{obtainedbyapplyingthe\|minimal operator\|nontrivial subnormal subgroup}}

	This is essentially the second equivalent formulation.		This is essentially the second equivalent formulation.

Vipul at 16:08, 9 May 2008

2008-05-09T16:08:35Z

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	{{group-subgroup property conjunction\|[[normal subgroup]]\|[[simple group]]}}		{{group-subgroup property conjunction\|normal subgroup\|simple group}}

	{{semibasicdef}}		{{semibasicdef}}

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2008-05-08T00:13:41Z

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{{group-subgroup property conjunction|[[normal subgroup]]|[[simple group]]}}

{{semibasicdef}}

==Definition==

===Symbol-free definition===

A [[subgroup]] of a [[group]] is termed a '''simple subnormal subgroup''' or a '''minimal subnormal subgroup''' if it satisfies the following equivalent conditions:

* It is [[simple group|simple as an abstract group]] and [[subnormal subgroup|subnormal as a subgroup]]
* It is ''minimal'' among subnormal subgroups, viz there is no smaller nontrivial subgroup of it that is subnormal in the whole group

{{obtainedbyapplyingthe|minimal operator|subnormality}}

This is essentially the second equivalent formulation.

===Equivalence of definitions===

{{proofat|[[Equivalence of definitions of simple subnormal subgroup]]}}

==Relation with other properties==

===Stronger properties===

* [[Simple normal subgroup]]

===Weaker properties===

* [[Component]]
* [[Subnormal subgroup]]

==Metaproperties==

{{transitive}}

In fact, the only simple subnormal subgroup of a simple subnormal subgroup is itself.