Group satisfying normalizer condition

Symbol-free definition
A group is said to satisfy the normalizer condition, if it satisfies the following equivalent conditions:


 * The normalizer of any proper subgroup properly contains it
 * There is no proper self-normalizing subgroup
 * Every subgroup is ascendant

Definition with symbols
A group $$G$$ is said to satisfy a normalizer condition if for any proper subgroup $$H$$ of $$G$$, $$H < N_G(H)$$ with the inclusion being strict (that is, $$H$$ is properly contained in its normalizer).

Groups satisfying the normalizer condition have been termed N-groups but the term N-group is also used for groups with a particular condition on normalizers of solvable subgroups.

Stronger properties

 * Weaker than::Nilpotent group: It turns out that for a finitely generated group, the two properties are equivalent.
 * Weaker than::Group in which every subgroup is subnormal

Weaker properties

 * Stronger than::Gruenberg group
 * Stronger than::Locally nilpotent group
 * Stronger than::Group having no proper abnormal subgroup
 * Stronger than::Group in which every maximal subgroup is normal