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
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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