# Difference between revisions of "Paranormal implies polynormal"

(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) |
||

Line 15: | Line 15: | ||

{{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>. | + | 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=== | ||

Line 21: | Line 21: | ||

{{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>. | + | 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

## Contents

## Statement

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

## Definitions used

For these definitions, denotes the conjugate of by , using the right-action convention (the action convention doesn't really matter). For subgroups , is the smallest subgroup of containing and invariant under the action of by conjugation.

### Paranormal subgroup

`Further information: Paranormal subgroup`

A subgroup of a group is termed **paranormal** in if, for any , is a contranormal subgroup of .

### Polynormal subgroup

`Further information: Polynormal subgroup`

A subgroup of a group is termed **polynormal** in if, for any , is a contranormal subgroup of .

## Facts used

## Proof

**Given**: A paranormal subgroup of a group .

**To prove**: For any , is contranormal in .

**Proof**: Clearly, is generated by for all , which in turn means that it is generated by the subgroups . is contranormal in each of these by definition of paranormality, so by fact (1), is contranormal in .