# Nilpotent Hall implies isomorph-conjugate

From Groupprops

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.

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## 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).