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