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 H of a finite group G 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 K of G whose order divides the order of H is contained in some conjugate of H.
  • It is a \pi-subgroup and is \pi-dominating for some set of primes \pi: In other words, H is a \pi-subgroup of G and every \pi-subgroup of G is contained in some conjugate of H.

Equivalence of definitions

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