# Focal subgroup of a subgroup

## Definition

Let be a subgroup of a group . We define the **focal subgroup** of in the following equivalent ways:

- The subgroup generated by the left quotients of pairs of elements of which are conjugate in .
- The subgroup generated by the right quotients of pairs of elements of which are conjugate in .

We use the notation or for the focal subgroup of in .

Note that the focal subgroup of in is contained within the commutator , and contains the commutator . In fact, we have the following string of inequalities:

.

## Facts

- A subgroup for which is termed a subgroup whose focal subgroup equals its commutator subgroup. Any conjugacy-closed subgroup has this property.
`Further information: Conjugacy-closed implies focal subgroup equals commutator subgroup` - A subgroup for which is termed a subgroup whose focal subgroup equals its intersection with the commutator subgroup. Any Sylow subgroup has this property.
`Further information: Focal subgroup theorem`