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 Hg is also equal to HK. 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.

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Paranormal subgroup. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Paranormal_subgroup&oldid=199305"