Question:Normal subgroup product with subgroups

Q: '''The product of a normal subgroup with any subgroup is a subgroup. Are normal subgroups the only subgroups with this property?'''

A: No. A subgroup whose product with any subgroup is a subgroup is termed a answer references::permutable subgroup (or quasinormal subgroup), i.e., it permutes with every subgroup. normal subgroups are permutable but answer references::permutable not implies normal. For instance, any subgroup of the Baer norm (the intersection of normalizers of all subgroups) is permutable but it need not be normal.