# Order-dominating Hall subgroup

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

View other subgroup property conjunctions | view all subgroup properties

## Definition

### Definition with symbols

A subgroup of a finite group is termed an **order-dominating Hall subgroup** if it satisfies the following equivalent conditions:

- It is both an order-dominating subgroup and a Hall subgroup: in other words, it is a Hall subgroup such that any subgroup of whose order divides the order of is contained in some conjugate of .
- It is a -subgroup and is -dominating for some set of primes : In other words, is a -subgroup of and every -subgroup of is contained in some conjugate of .

### Equivalence of definitions

`For full proof, refer: Pi-dominating pi-subgroup implies pi-Hall`