Definition:Abnormal Subgroup

Definition

Let $G$ be a group.

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

Then $H$ is an abnormal subgroup if and only if:

$\forall g \in G: g \in \gen {H, H^g}$

where $\gen {H, H^g}$ 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

• Results about abnormal subgroups can be found here.