Paranormal subgroup

In mathematics, in the field of group theory, a paranormal subgroup is a subgroup such that the subgroup generated by it and any conjugate of it, is also generated by it and a conjugate of it within that subgroup. In symbols, is paranormal in if given any in , the subgroup generated by and is also equal to . Equivalently, a subgroup is paranormal if its weak closure and normal closure coincide in all intermediate subgroups. Here are some facts relating paranormality to other subgroup properties: Every pronormal subgroup, and hence, every normal subgroup and every abnormal subgroup, is paranormal. Every paranormal subgroup is a polynormal subgroup. In finite solvable groups, every polynormal subgroup is paranormal.