Definition:Paranormal Subgroup

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Definition

Let $G$ be a group.

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

Then $H$ is a paranormal subgroup in $G$ if and only if the subgroup generated by $H$ and any conjugate of $H$ is also generated by $H$ and a conjugate of $H$ within that generated subgroup.

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

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

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 $g$.

Equivalently, a subgroup is paranormal if and only if its weak closure and normal closure coincide in all intermediate subgroup.

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:Polynormal Subgroup

• Pronormal Subgroup is Paranormal Subgroup
• Normal Subgroup is Paranormal Subgroup
• Abnormal Subgroup is Paranormal Subgroup
• Paranormal Subgroup is Polynormal Subgroup
• Polynormal Subgroup of Finite Solvable Group is Paranormal Subgroup

• Results about paranormal subgroups can be found here.