A conjugacy criterion for Hall subgroups in finite groups

TL;DR

This paper establishes a criterion based on a normal series for determining when a finite group has a unique conjugacy class of -Hall subgroups, advancing understanding of subgroup conjugacy properties.

Contribution

It provides a new criterion involving a normal series for when a finite group satisfies the conjugacy condition for -Hall subgroups, which was previously not well-characterized.

Findings

Criterion expressed via a normal series for -Hall subgroup conjugacy

Characterization of groups with a single conjugacy class of -Hall subgroups

Enhanced understanding of subgroup conjugacy in finite groups

Abstract

A finite group $G$ is said to satisfy $C_\pi$ for a set of primes $\pi$, if $G$ possesses exactly one class of conjugate $\pi$-Hall subgroups. In the paper we obtain a criterion for a finite group $G$ to satisfy $C_\pi$ in terms of a normal series of the group.