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.

External links

Kantor, William M.; Martino, Lino Di (12 January 1995). Groups of Lie Type and Their Geometries. Cambridge University Press. pp. 257–259. ISBN 9780521467902. https://books.google.com/books?id=iXXAZ3dmkNwC&dq=Paranormal+subgroup&pg=PA258.