# Hughes subgroup

From Groupprops

Template:Prime-parametrized subgroup-defining function

## Definition

Let be a prime and any group. The **Hughes subgroup** of , denoted , is defined as the smallest subgroup outside which all elements have order . In other words:

## Facts

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