Permutably complemented Hall subgroup

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: permutably complemented subgroup and Hall subgroup
View other subgroup property conjunctions | view all subgroup properties

Definition

A subgroup $H$ of a finite group $G$ is termed a permutably complemented Hall subgroup if it satisfies the following two conditions:

• $H$ is a Hall subgroup of $G$: the order and index of $H$ are relatively prime.
• $H$ is a permutably complemented subgroup of $G$: there exists a subgroup $K$ of $G$ such that $H\cap K$ is trivial and $HK=G$ (in other words, $H$ and $K$ are permutable complements). Note that $K$ also must be a Hall subgroup of $G$: if $H$ is $\pi$-Hall, $K$ is $\pi '$-Hall.

Relation with other properties

Stronger properties

• Hall retract
• Normal Hall subgroup: For full proof, refer: Normal Hall implies permutably complemented
• Hall direct factor

Weaker properties

• Lattice-complemented Hall subgroup