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