Normal Sylow subgroup: Difference between revisions
(New page: {{subgroup property conjunction|normal subgroup|Sylow subgroup}} ==Definition== A subgroup of a defining ingredient::finite group is termed a '''normal Sylow subgroup''' if it sa...) |
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==Relation with other properties== | ==Relation with other properties== | ||
===Stronger properties=== | |||
* [[Weaker than::Sylow direct factor]] | |||
===Weaker properties=== | ===Weaker properties=== | ||
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* [[Stronger than::Image-closed fully characteristic subgroup]] | * [[Stronger than::Image-closed fully characteristic subgroup]] | ||
* [[Stronger than::Normal subgroup]] | * [[Stronger than::Normal subgroup]] | ||
==Metaproperties== | |||
{{transfercondn}} | |||
{{intsubcondn}} | |||
{{imagecondn}} | |||
Revision as of 13:49, 28 September 2008
This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: normal subgroup and Sylow subgroup
View other subgroup property conjunctions | view all subgroup properties
Definition
A subgroup of a finite group is termed a normal Sylow subgroup if it satisfies the following equivalent conditions:
- It is a Sylow subgroup, and is normal in the whole group.
- It is a Sylow subgroup, and is subnormal in the whole group.
- It is a Sylow subgroup, and is characteristic in the whole group.
- It is a Sylow subgroup, and is fully characteristic in the whole group.
Relation with other properties
Stronger properties
Weaker properties
- Nilpotent normal subgroup
- Nilpotent characteristic subgroup
- Normal Hall subgroup
- Complemented normal subgroup
- Fully characteristic subgroup
- Characteristic subgroup
- Intermediately characteristic subgroup
- Isomorph-free subgroup
- Intermediately fully characteristic subgroup
- Image-closed characteristic subgroup
- Image-closed fully characteristic subgroup
- Normal subgroup
Metaproperties
Transfer condition
YES: This subgroup property satisfies the transfer condition: if a subgroup has the property in the whole group, its intersection with any subgroup has the property in that subgroup.
View other subgroup properties satisfying the transfer condition
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
Image condition
YES: This subgroup property satisfies the image condition, i.e., under any surjective homomorphism, the image of a subgroup satisfying the property also satisfies the property
View other subgroup properties satisfying image condition