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

TL;DR

This paper proves that in finite groups where maximal $ extit{ extbf{ extpi}}$-subgroups are conjugate, overgroups of $ extit{ extbf{ extpi}}$-Hall subgroups also inherit this property, confirming a long-standing problem.

Contribution

It establishes that overgroups of $ extit{ extbf{ extpi}}$-Hall subgroups in $ extit{ extbf{ extpi}}$-conjugate groups are also $ extit{ extbf{ extpi}}$-groups, solving Problem 17.44(b) from Kourovka notebook.

Findings

Overgroups of $ extit{ extbf{ extpi}}$-Hall subgroups are $ extit{ extbf{ extpi}}$-groups.

Confirmed inheritance of the $ extit{ extbf{ extpi}}$-property in finite groups.

Resolved a specific open problem in group theory.

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.