# Order-dominating Hall subgroup

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Revision as of 16:00, 20 February 2009 by Vipul (talk | contribs) (New page: {{wikilocal}} {{subgroup property conjunction|order-dominating subgroup|Hall subgroup}} ==Definition== ===Definition with symbols=== A subgroup <math>H</math> of a finite group <...)

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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

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## 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`