Subgroup of abelian group - Revision history

Vipul at 02:34, 16 February 2013

2013-02-16T02:34:00Z

← Older revision		Revision as of 02:34, 16 February 2013
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	==Relation with other properties==		==Relation with other properties==

			===Stronger properties===

			{\| class="sortable" border="1"
			! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
			\|-
			\| [[Weaker than::subgroup of cyclic group]] \|\| the whole group is [[cyclic group\|cyclic]] \|\| \|\| \|\| {{intermediate notions short\|subgroup of abelian group\|subgroup of cyclic group}}
			\|}
	===Weaker properties===		===Weaker properties===

	Properties weaker than this are properties true for all subgroups of abelian groups~~. These can be viewed at [[:Category:Abelian-tautological subgroup properties]]~~.		Properties weaker than this are properties true for all subgroups of abelian groups.

	* [[Stronger than::~~Central~~ subgroup]]~~: Also related:~~		{\| class="sortable" border="1"
	** [[Stronger than::~~Central~~ factor]]		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	** [[Stronger than::~~Transitively~~ normal subgroup]]		\|-
	** [[Stronger than::~~Normal~~ subgroup]]		\| [[Stronger than::central subgroup]] \|\| contained in the [[center]] \|\| \|\| \|\| {{intermediate notions short\|central subgroup\|subgroup of abelian group}}
	** [[Stronger than::~~Abelian~~ normal subgroup]]		\|-
			\| [[Stronger than::central factor]] \|\| product with centralizer is whole group \|\| (via central subgroup) \|\| (via central subgroup)\|\| {{intermediate notions short\|central factor\|subgroup of abelian group}}
			\|-
			\| [[Stronger than::transitively normal subgroup]] \|\| every normal subgroup of it is normal in the whole group. \|\| (via central factor)\|\| (via central factor)\|\| {{intermediate notions short\|transitively normal subgroup\|subgroup of abelian group}}
			\|-
			\| [[Stronger than::normal subgroup]] \|\| \|\| [[abelian implies every subgroup is normal]] \|\| any non-abelian group as a subgroup of itself. \|\| {{intermediate notions short\|normal subgroup\|subgroup of abelian group}}
			\|-
			\| [[Stronger than::abelian normal subgroup]] \|\| the subgroup is abelian as a group and is [[normal subgroup\|normal]] in the whole group. \|\| (via central subgroup) \|\| (via central subgroup) \|\| {{intermediate notions short\|abelian normal subgroup\|subgroup of abelian group}}
			\|}

Vipul: /* Relation with other properties */

2009-03-27T11:55:35Z

Relation with other properties

← Older revision		Revision as of 11:55, 27 March 2009
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	===Weaker properties===		===Weaker properties===

	Properties weaker than this are properties true for all subgroups of abelian groups. These can be viewed at [[:Category:~~Subgroup~~ properties ~~tautological for abelian groups~~]].		Properties weaker than this are properties true for all subgroups of abelian groups. These can be viewed at [[:Category:Abelian-tautological subgroup properties]].

	* [[Stronger than::Central subgroup]]: Also related:		* [[Stronger than::Central subgroup]]: Also related:

Vipul: Created page with '{{in-group subgroup property|subgroup|abelian group}} ==Definition== A subgroup of a group is termed a '''subgroup of abelian group''' if the ambient group is an [[abelian grou...'

2009-03-27T11:54:56Z

Created page with '{{in-group subgroup property|subgroup|abelian group}} ==Definition== A subgroup of a group is termed a '''subgroup of abelian group''' if the ambient group is an [[abelian grou...'

New page

{{in-group subgroup property|subgroup|abelian group}}

==Definition==

A subgroup of a group is termed a '''subgroup of abelian group''' if the ambient group is an [[abelian group]].

==Relation with other properties==

===Weaker properties===

Properties weaker than this are properties true for all subgroups of abelian groups. These can be viewed at [[:Category:Subgroup properties tautological for abelian groups]].

* [[Stronger than::Central subgroup]]: Also related:
** [[Stronger than::Central factor]]
** [[Stronger than::Transitively normal subgroup]]
** [[Stronger than::Normal subgroup]]
** [[Stronger than::Abelian normal subgroup]]