# Hughes subgroup

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:
$H_p(G) = \langle x \mid x^p \ne e \rangle$
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.