The existence of pronormal $\pi$-Hall subgroups in $E_{\pi }$-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 $⟨H,H^g⟩$. 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~$G$.