Definition:Paranormal Subgroup

Definition
Let $G$ be a group.

Let $H$ be a subgroup of $G$.

Then $H$ is a paranormal subgroup in $G$ iff the subgroup generated by $H$ and any conjugate of $H$ is also generated by $H$ and a conjugate of $H$ within that generated subgroup.

That is, $H$ is paranormal in $G$ iff:
 * $\forall g \in G: \exists k \in \left\langle{H, H^g}\right\rangle: \left\langle{H, H^k}\right\rangle = \left\langle{H, H^g}\right\rangle$

where:
 * $\left\langle{H, H^g}\right\rangle$ is the subgroup generated by $H$ and $H^g$
 * $H^g$ is the conjugate of $H$ by $g$.

Equivalently, a subgroup is paranormal if its weak closure and normal closure coincide in all intermediate subgroups.

Also see

 * Definition:Normal Subgroup
 * Definition:Subnormal Subgroup
 * Definition:Abnormal Subgroup
 * Definition:Weakly Abnormal Subgroup
 * Definition:Contranormal Subgroup
 * Definition:Self-Normalizing Subgroup
 * Definition:Pronormal Subgroup
 * Definition:Weakly Pronormal Subgroup
 * Definition:Polynormal Subgroup


 * Pronormal Subgroup is Paranormal Subgroup
 * Normal Subgroup is Paranormal Subgroup
 * Abnormal Subgroup is Paranormal Subgroup
 * Paranormal Subgroup is Polynormal Subgroup
 * Polynormal Subgroup of Finite Solvable Group is Paranormal Subgroup