\begin{definition}[Definition:Paranormal Subgroup]
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''' {{iff}} its weak closure and normal closure coincide in all intermediate subgroup.
\end{definition}