# Definition:Weakly Abnormal Subgroup

## Definition

Let $G$ be a group.

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

### Definition 1

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

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

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 $g$.

### Definition 2

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

if $H \le K \le G$, then $K$ is a self-normalizing subgroup of $G$

where $H \le K$ denotes that $H$ is a subgroup of $K$.

### Definition 3

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

if $H \le K \le G$, then $K$ is a contranormal subgroup of $G$

where $H \le K$ denotes that $H$ is a subgroup of $K$.