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.