Definition:Paranormal Subgroup

From ProofWiki
Jump to navigation Jump to search



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


  • Results about paranormal subgroups can be found here.