A Generalization of the Hughes Subgroup

TL;DR

This paper explores the properties of a generalized Hughes subgroup in finite groups, revealing that most groups exhibit only a few specific subgroup structures, with special cases involving Frobenius groups and nonabelian $q$-groups.

Contribution

It introduces a broader framework for Hughes subgroups, characterizing their structure in various finite groups and identifying unique cases involving Frobenius groups.

Findings

Most groups have $H_{\pi}(G)$ equal to 1, G, or a Hughes subgroup of a prime order.

Frobenius groups with specific kernel and complement structures are key exceptions.

Restrictions on primes $p$ and $q$ are established for these subgroup configurations.

Abstract

Let $G$ be a finite group, $\pi$ be a set of primes, and define $H_{\pi}(G)$ to be the subgroup generated by all elements of $G$ which do not have prime order for every prime in $\pi$. In this paper, we investigate some basic properties of $H_{\pi}(G)$ and its relationship to the Hughes subgroup. We show that for most groups, only one of three possibilities occur: $H_{\pi}(G) = 1$, $H_{\pi}(G)=G$, or $H_{\pi}(G) = H_{p}(G)$ for some prime $p \in \pi$. There is only one other possibility: $G$ is a Frobenius group whose Frobenius complement has prime order $p$, and whose Frobenius kernel, $F$, is a nonabelian $q$-group such that $H_{\pi}(G)$ arises as a proper and nontrivial Hughes subgroup of $F$. We investigate a few restrictions on the possible choices of the primes $p$ and $q$.