# Definition:Abnormal Subgroup

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## 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**.