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, formula_1 is paranormal in formula_2 if given any formula_3 in formula_2, the subgroup formula_5 generated by formula_1 and formula_7 is also equal to formula_8. 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: