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