# Definition:Weakly Pronormal Subgroup

## Definition

Let $G$ be a group.

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

### Definition 1

$H$ is weakly pronormal in $G$ if and only if:

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

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 $x$.

### Definition 2

$H$ is weakly pronormal in $G$ if and only if:

if $H \le K \le L \le G$ are such that $K$ is a normal subgroup of $L$, then $K N_L \left({H}\right) = L$

where:

$H \le K$ denotes that $H$ is a subgroup of $K$
$N_L \left({H}\right)$ denotes the normalizer of $H$ in $L$.