Hughes subgroup

Definition

Let $p$ be a prime and $G$ any group. The Hughes subgroup of $G$, denoted $H_{p}(G)$, is defined as the smallest subgroup outside which all elements have order $p$. In other words:

$H_{p}(G)=\langle x\mid x^{p}\neq e\rangle$

Facts

The Hughes conjecture states that for a finite group, the Hughes subgroup is either trivial, or the whole group, or has index $p$. This conjecture is now known to be false. A group for which this conjecture is true is termed a group satisfying Hughes conjecture.