# Order-conjugate Hall subgroup

From Groupprops

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: order-conjugate subgroup and Hall subgroup

View other subgroup property conjunctions | view all subgroup properties

## Definition

A subgroup of a group is termed an **order-conjugate Hall subgroup** if it satisfies the following two conditions:

- It is a Hall subgroup: its order and index are relatively prime.
- It is an order-conjugate subgroup: it is conjugate to any subgroup of the same order.

## Relation with other properties

### Stronger properties

- Sylow subgroup
- Normal Hall subgroup
- Order-dominating Hall subgroup:
*For proof of the implication, refer order-dominating implies order-conjugate and for proof of its strictness (i.e. the reverse implication being false) refer Order-conjugate and Hall not implies order-dominating*.