Weakly abnormal subgroup

Symbol-free definition
A subgroup of a group is termed weakly abnormal or upward-closed self-normalizing or intermediately contranormal if it satisfies the following equivalent conditions:


 * 1) Every element of the group lies in the closure of this subgroup under the action by conjugation by the cyclic subgroup generated by that element.
 * 2) Every subgroup containing that subgroup is a self-normalizing subgroup of the whole group.
 * 3) The subgroup is a contranormal subgroup in every intermediate subgroup.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed weakly abnormal or upward-closed self-normalizing or intermediately contranormal if it satisfies the following equivalent conditions:


 * 1) Given any $$g \in G$$, $$g \in H^{\langle g \rangle}$$. Here $$H^{\langle g \rangle}$$ is the smallest subgroup of $$G$$ containing $$H$$, which is closed under the action by conjugation by the cyclic subgroup generated by $$g$$
 * 2) If $$H \le K \le G$$, then $$K$$ is a defining ingredient::self-normalizing subgroup of $$G$$.
 * 3) If $$H \le L \le G$$, then $$H$$ is a defining ingredient::contranormal subgroup of $$L$$.

Formalisms
$$H$$ is weakly abnormal in $$G$$ if and only if every subgroup of $$G$$ containing $$H$$ is self-normalizing.

$$H$$ is weakly abnormal in $$G$$ if and only if $$H$$ is contranormal in every intermediate subgroup.

Stronger properties

 * Non-normal maximal subgroup
 * Weaker than::Abnormal subgroup

Weaker properties

 * Stronger than::Weakly pronormal subgroup
 * Stronger than::Self-normalizing subgroup
 * Stronger than::Contranormal subgroup
 * Stronger than::Paracharacteristic subgroup
 * Stronger than::Paranormal subgroup
 * Stronger than::Polycharacteristic subgroup
 * Stronger than::Polynormal subgroup

Metaproperties
If a subgroup is weakly abnormal in the whole group, it is also weakly abnormal in every intermediate subgroup.

If a subgroup is weakly abnormal in the whole group, then every subgroup containing it is also weakly abnormal in the whole group.