The Hall property ${\mathcal{D}}_{\pi }$ is inherited by overgroups of $\pi$-Hall subgroups

Abstract

Let $\pi$ be a set of primes. We say that a finite group $G$ is a ${\mathcal{D}}_{\pi }$-group if the maximal $\pi$-subgroups of $G$ are conjugate. In this paper, we give an affirmative answer to Problem 17.44(b) from "Kourovka notebook", namely we prove that in a ${\mathcal{D}}_{\pi }$-group an overgroup of a $\pi$-Hall subgroup is always a ${\mathcal{D}}_{\pi }$-group.

arXiv admin note: text overlap with arXiv:1504.03137