# Category:Pronormal Subgroups

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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 $G$** iff 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 $G$** iff:

- $\forall g \in G: \exists k \in \left\langle{H, H^g}\right\rangle: H^k = H^g$

where:

- $\left\langle{H, H^g}\right\rangle$ is the subgroup generated by $H$ and $H^g$
- $H^g$ is the conjugate of $H$ by $g$.

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