Definition:Pronormal Subgroup
Jump to navigation
Jump to search
Definition
Let $G$ be a group.
Let $H$ be a subgroup of $G$.
Then $H$ is a pronormal subgroup in $G$ if and only if each of its conjugates in $G$ is conjugate to it already in the subgroup generated by $H$ and its conjugate.
That is, $H$ is pronormal in $G$ if and only if:
- $\forall g \in G: \exists k \in \gen {H, H^g}: H^k = H^g$
where:
- $\gen {H, H^g}$ is the subgroup generated by $H$ and $H^g$
- $H^g$ is the conjugate of $H$ by $g$.
Also see
- Definition:Normal Subgroup
- Definition:Subnormal Subgroup
- Definition:Abnormal Subgroup
- Definition:Weakly Abnormal Subgroup
- Definition:Contranormal Subgroup
- Definition:Self-Normalizing Subgroup
- Definition:Weakly Pronormal Subgroup
- Definition:Paranormal Subgroup
- Definition:Polynormal Subgroup
- Normal Subgroup is Pronormal Subgroup
- Sylow Subgroup is Pronormal Subgroup
- Pronormal Subnormal Subgroup is Normal Subgroup
- Abnormal Subgroup is Pronormal Subgroup
- Results about pronormal subgroups can be found here.