# Difference between revisions of "Nilpotent Hall implies isomorph-conjugate"

From Groupprops

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==Facts used== | ==Facts used== | ||

− | [[Nilpotent Hall subgroups of same order are conjugate]] | + | # [[uses::Nilpotent Hall subgroups of same order are conjugate]] |

− | + | # [[uses::Sylow implies intermediately isomorph-conjugate]] | |

+ | # [[uses::Hall implies join of Sylow subgroups]] | ||

+ | # [[uses::Nilpotent join of intermediately isomorph-conjugate subgroups is intermediately isomorph-conjugate]] | ||

==Related facts== | ==Related facts== | ||

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* [[Hall not implies isomorph-automorphic]] | * [[Hall not implies isomorph-automorphic]] | ||

* [[Nilpotent Hall not implies order-conjugate]] | * [[Nilpotent Hall not implies order-conjugate]] | ||

+ | |||

+ | ==Proof== | ||

+ | |||

+ | ===Proof using fact (1)=== | ||

+ | |||

+ | The proof follows directly from fact (1). | ||

+ | |||

+ | ===Proof using facts (2)-(4)=== | ||

+ | |||

+ | The proof follows directly by combining facts (2)-(4). |

## Latest revision as of 10:19, 29 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., Nilpotent Hall subgroup (?)) must also satisfy the second subgroup property (i.e., Isomorph-conjugate subgroup (?)). In other words, every nilpotent Hall subgroup of finite group is a isomorph-conjugate 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

## Contents

## Statement

Any Nilpotent Hall subgroup (?) (i.e., a Hall subgroup (?) that is also a Nilpotent group (?))of a finite group is isomorph-conjugate: it is conjugate to any isomorphic subgroup.

## Facts used

- Nilpotent Hall subgroups of same order are conjugate
- Sylow implies intermediately isomorph-conjugate
- Hall implies join of Sylow subgroups
- Nilpotent join of intermediately isomorph-conjugate subgroups is intermediately isomorph-conjugate

## Related facts

- Hall not implies isomorph-conjugate
- Hall not implies isomorph-automorphic
- Nilpotent Hall not implies order-conjugate

## Proof

### Proof using fact (1)

The proof follows directly from fact (1).

### Proof using facts (2)-(4)

The proof follows directly by combining facts (2)-(4).