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, 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.
