# Category:Pronormal Subgroups

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 $g$.

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