Difference between revisions of "Sylow implies pronormal"

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(New page: {{subgroup property implication in| group property = finite group| stronger = Sylow subgroup| weaker = pronormal subgroup}} ==Statement== Any Sylow subgroup of a finite group is ...)
 
(Corollaries)
 
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* [[Sylow normalizer implies abnormal]]
 
* [[Sylow normalizer implies abnormal]]
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* [[Sylow normalizer implies upward-closed self-normalizing]]
 
* [[Sylow implies intermediately subnormal-to-normal]]
 
* [[Sylow implies intermediately subnormal-to-normal]]
  

Latest revision as of 19:30, 19 September 2008

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties, when the big group is a finite group. That is, it states that in a Finite group (?), every subgroup satisfying the first subgroup property (i.e., Sylow subgroup (?)) must also satisfy the second subgroup property (i.e., Pronormal subgroup (?)). In other words, every Sylow subgroup of finite group is a pronormal subgroup of finite group.
View all subgroup property implications in finite groups | View all subgroup property non-implications in finite groups | View all subgroup property implications | View all subgroup property non-implications

Statement

Any Sylow subgroup of a finite group is pronormal.

Definitions used

Sylow subgroup

Further information: Sylow subgroup

Pronormal subgroup

Further information: Pronormal subgroup

Related facts

Stronger facts

Corollaries

Facts used

  1. Sylow implies intermediately isomorph-conjugate
  2. Intermediately isomorph-conjugate implies pronormal

Proof

The proof follows directly by combining facts (1) and (2).