Definition:Weakly Pronormal Subgroup/Definition 1

Definition
Let $G$ be a group.

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

$H$ is weakly pronormal in $G$ :


 * $\forall g \in G: \exists x \in H^{\gen g}: H^x = H^g$

where:
 * $H^{\gen g}$ denotes the smallest subgroup of $G$ containing $H$, generated by the conjugacy action by the cyclic subgroup of $G$ generated by $g$
 * $H^x$ denotes the conjugate of $H$ by $x$.

Also see

 * Equivalence of Definitions of Weakly Pronormal Subgroup