Paranormal subgroup explained

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

Book: William Kantor. Kantor. William M.. Martino. Lino Di. Groups of Lie Type and Their Geometries. 12 January 1995. Cambridge University Press. 9780521467902. 257–259.