Definition:Polynormal Subgroup

Definition
Let $G$ be a group.

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

Then $H$ is a polynormal subgroup in $G$ iff the conjugate closure of $H$ by any element of $G$ can also be achieved via conjugate closure by some element in the subgroup generated.

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

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

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:Paranormal Subgroup


 * Weakly Pronormal Subgroup is Polynormal Subgroup
 * Paranormal Subgroup is Polynormal Subgroup
 * Polynormal Subgroup of Finite Solvable Group is Paranormal Subgroup