\begin{definition}[Definition:Polynormal Subgroup]
Let $G$ be a group.
Let $H$ be a subgroup of $G$.
Then $H$ is a '''polynormal subgroup''' of $G$ {{iff}} 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 $G$''' {{iff}}:
: $\forall g \in G: H^{\left\langle g \right\rangle} = H^{H^{\left\langle g \right\rangle}}$
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 $k \in H^{\left\langle g \right\rangle}$.
\end{definition}