Normal subset of a group

Definition
A subset of a group is termed a normal subset if it satisfies the following equivalent conditions:


 * 1) It is a union of conjugacy classes.
 * 2) It equals its conjugate by any element of the group.

This is a generalization, to arbitrary subsets of groups, of the notion of normal subgroup. Specifically, a subgroup is a normal subgroup if and only if it is a normal subset.