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, $$H$$ is paranormal in $$G$$ if given any $$g$$ in $$G$$, the subgroup $$K$$ generated by $$H$$ and $$H^g$$ is also equal to $$H^K$$. 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.