Definition:Pronormal Subgroup

Definition
Let $G$ be a group.

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

Then $H$ is a pronormal subgroup in $G$ iff each of its conjugates in $G$ is conjugate to it already in the subgroup generated by $H$ and its conjugate.

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

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 $g$.

Also see

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


 * Normal Subgroup is Pronormal Subgroup
 * Sylow Subgroup is Pronormal Subgroup
 * Pronormal Subnormal Subgroup is Normal Subgroup
 * Abnormal Subgroup is Pronormal Subgroup