On the intersections of solvable Hall subgroups in finite groups

TL;DR

This paper investigates a conjecture about the intersections of solvable Hall subgroups in finite groups, showing that minimal counterexamples are almost simple groups of Lie type.

Contribution

It proves that if a finite group has a solvable -Hall subgroup, then a specific intersection property holds, identifying the minimal counterexamples as almost simple groups of Lie type.

Findings

Counterexamples are almost simple groups of Lie type.

The conjecture holds under certain conditions for solvable -Hall subgroups.

Minimal counterexamples are characterized as almost simple groups of Lie type.

Abstract

In the paper we consider the following conjecture: if a finite group $G$ possesses a solvable $\pi$-Hall subgroup $H$, then there exist elements $x,y,z,t\in G$ such that the identity $H\cap H^x\cap H^y\cap H^z\cap H^t=O_\pi(G)$ holds. The minimal counter example is shown to be an almost simple group of Lie type.