# Hall not implies order-conjugate

From Groupprops

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., Hall subgroup) neednotsatisfy the second subgroup property (i.e., order-conjugate subgroup)

View a complete list of subgroup property non-implications | View a complete list of subgroup property implications

Get more facts about Hall subgroup|Get more facts about order-conjugate subgroup

EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property Hall subgroup but not order-conjugate subgroup|View examples of subgroups satisfying property Hall subgroup and order-conjugate subgroup

## Contents

## Statement

There can exist a finite group and Hall subgroups of of the same order that are not conjugate in .

## Related facts

### Similar facts about Hall subgroups

- Hall not implies automorph-conjugate
- Hall not implies isomorph-automorphic
- Hall not implies order-isomorphic
- Hall and order-conjugate not implies order-dominating
- Hall and order-conjugate not implies order-dominated
- Hall not implies procharacteristic
- Hall not implies pronormal

### Opposite facts about Hall subgroups

- Sylow implies order-dominating
- Sylow implies order-conjugate
- Sylow implies order-dominated
- Hall implies order-dominating in finite solvable
- Hall implies order-conjugate in finite solvable
- Hall implies order-dominated in finite solvable
- Nilpotent Hall subgroups of same order are conjugate
- Hall implies paracharacteristic
- Hall implies paranormal
- Hall implies intermediately subnormal-to-normal

## Facts used

- Hall not implies automorph-conjugate
- Hall not implies isomorph-automorphic
- Hall not implies order-isomorphic

## Proof

The proof follows from either of facts (1), (2) or (3). Fact (1) gives examples where the two Hall subgroups are automorphic subgroups but are not conjugate (in other words, there is an outer automorphism sending one Hall subgroup to the other, but no inner automorphism doing it). Fact (2) gives examples where the two Hall subgroups are isomorphic but there is no automorphism sending the first to the second. Fact (3) gives examples where the two subgroups have the same order but are not isomorphic, and hence cannot be conjugate.