Abnormal subgroup

In mathematics, in the field of group theory, an abnormal subgroup is a subgroup $H$ of a group $G$ such that for any $x$ in $G$, $x$ lies in the subgroup generated by $H$ and $H^x$.

Here are some facts relating abnormality to other subgroup properties:

* Every abnormal subgroup is a self-normalizing subgroup, as well as a contranormal subgroup.
* The only normal subgroup that is also abnormal is the whole group.
* Every abnormal subgroup is a weakly abnormal subgroup, and every weakly abnormal subgroup is a self-normalizing subgroup.
* Every abnormal subgroup is a pronormal subgroup, and hence a weakly pronormal subgroup, a paranormal subgroup, and a polynormal subgroup.