Conjugacy-closed Hall subgroup

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

Definition

Symbol-free definition

A subgroup of a group is termed a conjugacy-closed Hall subgroup if it satisfies the following conditions:

• It is conjugacy-closed: Any two elements of the subgroup that are conjugate in the whole group are conjugate in the subgroup.
• It is a Hall subgroup: Its order and index are relatively prime.

Relation with other properties

Stronger properties

• Sylow direct factor
• Sylow retract
• Hall direct factor
• Hall retract

Weaker properties

• Conjugacy-closed subgroup
• Hall subgroup