Definition:Polynormal Subgroup
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Definition
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 $k \in H^{\left\langle g \right\rangle}$.
Also see
- Definition:Normal Subgroup
- Definition:Subnormal Subgroup
- Definition:Abnormal Subgroup
- Definition:Weakly Abnormal Subgroup
- Definition:Contranormal Subgroup
- Definition:Self-Normalizing Subgroup
- Definition:Pronormal Subgroup
- Definition:Weakly Pronormal Subgroup
- Definition:Paranormal Subgroup
- Weakly Pronormal Subgroup is Polynormal Subgroup
- Paranormal Subgroup is Polynormal Subgroup
- Polynormal Subgroup of Finite Solvable Group is Paranormal Subgroup
- Results about polynormal subgroups can be found here.