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:

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

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.