# Category:Polynormal Subgroups

This category contains results about Polynormal Subgroups.

Let $G$ be a group.

Let $H$ be a subgroup of $G$.

Then $H$ is a polynormal subgroup of $G$ if and only if for all $g \in G$, the conjugate closure of $H$ in $\langle H, g \rangle$ is equal to the conjugate closure of $H$ in $H^{\langle g \rangle}$.

That is, $H$ is polynormal in $G$ if and only if:

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

where:

$H^{\left\langle g \right\rangle}$ is the subgroup generated by the set of all elements of the form $g^nhg^{-n}$ where $h \in H, n \in \mathbb{Z}$,
$H^{H^{\left\langle g \right\rangle}}$ is the subgroup generated by the set of all elements of the form $khk^{-1}$ where $k \in H^{\left\langle g \right\rangle}$.

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