Definition:Polynormal Subgroup

Definition
Let $G$ be a group.

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

Then $H$ is a polynormal subgroup of $G$ for all $g \in G$, the conjugate closure of $H$ in $\langle H, g \rangle$ is equal to the conjugate closure of $H$ in $H^{\langle g \rangle}$.

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

where:
 * $H^{\left\langle g \right\rangle}$ is the subgroup generated by the set of all elements of the form $g^nhg^{-n}$ where $h \in H, n \in \mathbb{Z}$,
 * $H^{H^{\left\langle g \right\rangle}}$ is the subgroup generated by the set of all elements of the form $khk^{-1}$ where $k \in H^{\left\langle g \right\rangle}$.

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