Pronormal Hall subgroup

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: pronormal 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 a pronormal Hall subgroup if it satisfies the following two conditions:

1. $H$ is a Hall subgroup of $G$: The order and index of $H$ in $G$ are relatively prime.
2. $H$ is a pronormal subgroup of $G$: If $K$ is a conjugate subgroup to $H$ in $G$, then $H, K$ are conjugate subgroups inside $\langle H, K \rangle$.

Relation with other properties

Stronger properties

• Sylow subgroup
• Intermediately isomorph-conjugate Hall subgroup: Also related:
  • Nilpotent Hall subgroup
  • Intermediately order-conjugate Hall subgroup
  • Normal Hall subgroup
  • Hall retract

Weaker properties

• Hall subgroup: For full proof, refer: Hall not implies pronormal
• Hall WNSCDIN-subgroup