Question:Normal subgroup product with subgroups

This question is about normal subgroup and product of subgroups.| See more questions about normal subgroup

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 permutable subgroup (or quasinormal subgroup), i.e., it permutes with every subgroup. normal subgroups are permutable but 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.