Subgroup properties

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, is paranormal in if given any in , the subgroup generated by and is also equal to . 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. (Wikipedia).

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Related pages

Abnormal subgroup | Subgroup | Mathematics | Pronormal subgroup | Polynormal subgroup | Generating set of a group | Group theory | Conjugacy class | Normal subgroup