Definition:Abnormal Subgroup

Definition
Let $G$ be a group.

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

Then $H$ is an abnormal subgroup iff:


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

where $\left\langle{H, H^g}\right\rangle$ is the subgroup of $G$ generated by $H$ and the conjugate of $H$ by $g$.

Also see

 * Definition:Normal Subgroup
 * Definition:Subnormal Subgroup
 * Definition:Weakly Abnormal Subgroup
 * Definition:Contranormal Subgroup
 * Definition:Self-Normalizing Subgroup
 * Definition:Pronormal Subgroup
 * Definition:Weakly Pronormal Subgroup
 * Definition:Paranormal Subgroup
 * Definition:Polynormal Subgroup


 * Abnormal Subgroup is Self-Normalizing Subgroup
 * Abnormal Subgroup is Contranormal Subgroup
 * Subgroup both Normal and Abnormal is Whole Group
 * Abnormal Subgroup is Weakly Abnormal Subgroup
 * Weakly Abnormal Subgroup is Self-Normalizing Subgroup
 * Abnormal Subgroup is Pronormal Subgroup
 * Abnormal Subgroup is Weakly Pronormal Subgroup
 * Abnormal Subgroup is Paranormal Subgroup
 * Abnormal Subgroup is Polynormal Subgroup