The existence of pronormal $\pi$-Hall subgroups in $E_\pi$-groups

TL;DR

This paper proves that in finite groups with a $ ext{ extpi}$-Hall subgroup, every normal subgroup also has a $ ext{ extpi}$-Hall subgroup that is pronormal, extending known subgroup conjugacy properties.

Contribution

It establishes the existence of pronormal $ ext{ extpi}$-Hall subgroups in all normal subgroups of such finite groups, a new result in subgroup theory.

Findings

Normal subgroups have pronormal $ ext{ extpi}$-Hall subgroups.

Extension of conjugacy properties to all normal subgroups.

Generalization of subgroup existence in finite groups.

Abstract

A subgroup $H$ of a group $G$ is called {\it pronormal}, if for every $g\in G$ subgroups $H$ and $H^g$ are conjugate in $\langle H, H^g\rangle$. It is proven that if a finite group $G$ possesses a $\pi$-Hall subgroup for a set of primes $\pi$, the every its normal subgroup (in particular, $G$ itself) possesses a $\pi$-Hall subgroup that is pronormal in~$G$.