Subgroup whose commutator with any subset is normal

Definition
A subgroup of a group is termed a subgroup whose commutator with any subset is normal if the commutator of that subgroup with any subset of the whole group is a normal subgroup of the whole group.

Stronger properties

 * Weaker than::Transitively normal subgroup:
 * Weaker than::Direct factor
 * Weaker than::Central factor