Weakly normal subgroup

Symbol-free definition
A subgroup of a group is termed weakly normal if whenever any conjugate of it is contained in its normalizer, the conjugate is actually contained in the subgroup itself. Equivalently, a subgroup is weakly normal if it is a defining ingredient::weakly closed subgroup inside its normalizer relative to the whole group.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed weakly normal if for any $$g$$ in $$G$$, if $$H^g \le N_G(H)$$ implies $$H^g \le H$$.

(Here $$H^g = g^{-1}Hg$$ denotes the conjugate of $$H$$ by $$g$$ under the right action. We can use either the left or the right action for the definition).

Stronger properties

 * Weaker than::Normal subgroup
 * Weaker than::Pronormal subgroup:
 * Weaker than::NE-subgroup:
 * Weaker than::Paranormal subgroup:
 * Weaker than::Self-normalizing subgroup
 * Weaker than::Weakly characteristic subgroup

Weaker properties

 * Stronger than::Intermediately subnormal-to-normal subgroup:
 * Stronger than::Subgroup with self-normalizing normalizer

Metaproperties
If $$H \le K \le G$$ are groups and $$H$$ is weakly normal in $$G$$, then $$H$$ is weakly normal in $$K$$.