Central subgroup of normalizer: Difference between revisions
(New page: {{group-subgroup property conjunction|WC-subgroup|Abelian group}} ==Definition== A subgroup of a group is termed a '''central subgroup of normalizer''' if it satisfies the follow...) |
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{{group-subgroup property conjunction| | {{group-subgroup property conjunction|central factor of normalizer|abelian group}} | ||
==Definition== | ==Definition== | ||
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* It is a [[defining ingredient::central subgroup]] (i.e., is contained in the [[defining ingredient::center]]) of its [[normalizer]]. | * It is a [[defining ingredient::central subgroup]] (i.e., is contained in the [[defining ingredient::center]]) of its [[normalizer]]. | ||
* It is [[Abelian group|Abelian]] and is a [[ | * It is [[Abelian group|Abelian]] and is a [[central factor of normalizer]]. | ||
* Any [[inner automorphism]] of the whole group that leaves the subgroup invariant, must act trivially on the subgroup. | * Any [[inner automorphism]] of the whole group that leaves the subgroup invariant, must act trivially on the subgroup. | ||
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{{obtainedbyapplyingthe|in-normalizer operator|central subgroup}} | {{obtainedbyapplyingthe|in-normalizer operator|central subgroup}} | ||
==Facts== | |||
* [[Burnside's normal p-complement theorem]]: This states that if a Sylow subgroup is a central subgroup of its normalizer, then it is a [[retract]], i.e., it possesses a normal complement. | |||
==Metaproperties== | ==Metaproperties== | ||
{{intsubcondn}} | {{intsubcondn}} | ||
Latest revision as of 21:50, 26 July 2009
This article describes a property that arises as the conjunction of a subgroup property: central factor of normalizer with a group property (itself viewed as a subgroup property): abelian group
View a complete list of such conjunctions
Definition
A subgroup of a group is termed a central subgroup of normalizer if it satisfies the following equivalent conditions:
- It is a central subgroup (i.e., is contained in the center) of its normalizer.
- It is Abelian and is a central factor of normalizer.
- Any inner automorphism of the whole group that leaves the subgroup invariant, must act trivially on the subgroup.
Formalisms
In terms of the in-normalizer operator
This property is obtained by applying the in-normalizer operator to the property: central subgroup
View other properties obtained by applying the in-normalizer operator
Facts
- Burnside's normal p-complement theorem: This states that if a Sylow subgroup is a central subgroup of its normalizer, then it is a retract, i.e., it possesses a normal complement.
Metaproperties
Intermediate subgroup condition
YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition