\begin{definition}[Definition:Pronormal Subgroup]
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}}:
: $\forall g \in G: \exists k \in \gen {H, H^g}: H^k = H^g$
where:
: $\gen {H, H^g}$ is the subgroup generated by $H$ and $H^g$
: $H^g$ is the conjugate of $H$ by $g$.
\end{definition}