Definition:Weakly Abnormal Subgroup
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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 $K$.
Also see
- Definition:Normal Subgroup
- Definition:Subnormal Subgroup
- Definition:Abnormal Subgroup
- Definition:Self-Normalizing Subgroup
- Definition:Contranormal Subgroup
- Definition:Pronormal Subgroup
- Definition:Weakly Pronormal Subgroup
- Definition:Paranormal Subgroup
- Definition:Polynormal Subgroup