Hughes subgroup

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Template:Prime-parametrized subgroup-defining function

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.