Serial subgroup - Revision history

Vipul at 22:09, 7 August 2009

2009-08-07T22:09:10Z

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	{{variation of\|normality}}		{{variation of\|normality}}
	{{~~variation of~~\|~~subnormality~~}}		{{finitarily equivalent to\|subnormal subgroup}}

	==Definition==		==Definition==

Vipul: /* Relation with other properties */

2009-08-07T21:55:11Z

Relation with other properties

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	===Stronger properties===		===Stronger properties===

	* [[Ascendant subgroup]]		* [[Weaker than::Ascendant subgroup]]
	* [[Descendant subgroup]]		* [[Weaker than::Descendant subgroup]]
	* [[Subnormal subgroup]]		* [[Weaker than::Subnormal subgroup]]
	* [[2-subnormal subgroup]]		* [[Weaker than::2-subnormal subgroup]]
	* [[Composition subgroup]]		* [[Weaker than::Composition subgroup]]

	===Related group properties===		===Related group properties===

Vipul at 21:54, 7 August 2009

2009-08-07T21:54:41Z

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	===Symbol-free definition===		===Symbol-free definition===

	~~A subgroup of a group is termed a '''serial subgroup''' if there is a [[normal series]] starting at the subgroup and ending at the whole group.~~		{{fillin}}

	===In terms of the serial closure operator===		===In terms of the serial closure operator===

Vipul at 21:54, 7 August 2009

2009-08-07T21:54:09Z

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	{{subgroup property}}		{{subgroup property}}

	{{~~variationof~~\|normality}}		{{variation of\|normality}}
			{{variation of\|subnormality}}

	==Definition==		==Definition==

Vipul: 1 revision

2008-05-08T00:13:07Z

1 revision

← Older revision	Revision as of 00:13, 8 May 2008
(No difference)

Vipul at 12:52, 27 February 2007

2007-02-27T12:52:31Z

New page

{{subgroup property}}

{{variationof|normality}}

==Definition==

===Symbol-free definition===

A subgroup of a group is termed a '''serial subgroup''' if there is a [[normal series]] starting at the subgroup and ending at the whole group.

===In terms of the serial closure operator===

The property of being a serial subgroup is obtained by applying the [[serial closure operator]] to the subgroup property of being [[normal subgroup|normal]].

==Relation with other properties==

===Stronger properties===

* [[Ascendant subgroup]]
* [[Descendant subgroup]]
* [[Subnormal subgroup]]
* [[2-subnormal subgroup]]
* [[Composition subgroup]]

===Related group properties===

* [[Absolutely simple group]] is a group that has no proper nontrivial serial subgroup. Thus, this property is obtained by applying the [[simple group operator]] to the subgroup property of being simple

==Metaproperties==

{{transitive}}

{{proofat|[[Serial satisfies transitivity]]}}

Since any [[serially closed subgroup property]] is transitive, the property of being a serial subgroup is a [[transitive subgroup property]].

{{trim}}

The property of being a serial subgroup is [[trim subgroup property|trim]], because the property of being normal is itself trim.