Difference between revisions of "Paranormal implies polynormal"

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(New page: {{subgroup property implication| stronger = paranormal subgroup| weaker = polynormal subgroup}} ==Statement== Any paranormal subgroup of a group is also a polynormal subgroup. =...)
 
(Definitions used)
 
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{{further|[[Paranormal subgroup]]}}
 
{{further|[[Paranormal subgroup]]}}
  
A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''paranormal''' in <math>G</math> if <math>H</math> is a [[contranormal subgroup]] of <math>\langle H, H^g \rangle</math>.
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A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''paranormal''' in <math>G</math> if, for any <math>g \in G</math>, <math>H</math> is a [[contranormal subgroup]] of <math>\langle H, H^g \rangle</math>.
  
 
===Polynormal subgroup===
 
===Polynormal subgroup===
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{{further|[[Polynormal subgroup]]}}
 
{{further|[[Polynormal subgroup]]}}
  
A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''polynormal''' in <math>G</math> if <math>H</math> is a [[contranormal subgroup]] of <math>H^{\langle g \rangle}</math>.
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A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''polynormal''' in <math>G</math> if, for any <math>g \in G</math>, <math>H</math> is a [[contranormal subgroup]] of <math>H^{\langle g \rangle}</math>.
  
 
==Facts used==
 
==Facts used==

Latest revision as of 20:59, 24 September 2008

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., paranormal subgroup) must also satisfy the second subgroup property (i.e., polynormal subgroup)
View all subgroup property implications | View all subgroup property non-implications
Get more facts about paranormal subgroup|Get more facts about polynormal subgroup

Statement

Any paranormal subgroup of a group is also a polynormal subgroup.

Definitions used

For these definitions, H^g = g^{-1}Hg denotes the conjugate of H by g, using the right-action convention (the action convention doesn't really matter). For subgroups H,K \le G, H^K is the smallest subgroup of G containing H and invariant under the action of K by conjugation.

Paranormal subgroup

Further information: Paranormal subgroup

A subgroup H of a group G is termed paranormal in G if, for any g \in G, H is a contranormal subgroup of \langle H, H^g \rangle.

Polynormal subgroup

Further information: Polynormal subgroup

A subgroup H of a group G is termed polynormal in G if, for any g \in G, H is a contranormal subgroup of H^{\langle g \rangle}.

Facts used

  1. Contranormality is upper join-closed

Proof

Given: A paranormal subgroup H of a group G.

To prove: For any g \in G, H is contranormal in H^{\langle g \rangle}.

Proof: Clearly, H^{\langle g \rangle} is generated by H^k for all k \in \langle g \rangle, which in turn means that it is generated by the subgroups \langle H, H^k \rangle. H is contranormal in each of these by definition of paranormality, so by fact (1), H is contranormal in H^{\langle g \rangle}.