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