# Category:Pronormal Subgroups

Jump to navigation
Jump to search

This category contains results about **Pronormal Subgroups**.

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 $g$.

*This category currently contains no pages or media.*