Pronormal Hall subgroup

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 $$\langle H, K \rangle$$.

Stronger properties

 * Weaker than::Sylow subgroup
 * Weaker than::Intermediately isomorph-conjugate Hall subgroup: Also related:
 * Weaker than::Nilpotent Hall subgroup
 * Weaker than::Intermediately order-conjugate Hall subgroup
 * Weaker than::Normal Hall subgroup
 * Weaker than::Hall retract

Weaker properties

 * Stronger than::Hall subgroup:
 * Stronger than::Hall WNSCDIN-subgroup