Intermediately normal-to-characteristic subgroup

Symbol-free definition
A subgroup of a group is termed intermediately normal-to-characteristic if it satisfies the following equivalent conditions:


 * Whenever it is normal in any intermediate subgroup, it is also characteristic in the intermediate subgroup.
 * It is an defining ingredient::intermediately characteristic subgroup in its defining ingredient::normalizer.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed intermediately normal-to-characteristic in $$G$$ if it satisfies the following equivalent conditions:


 * For any subgroup $$K$$ of $$G$$ containing $$H$$ such that $$H$$ is normal in $$K$$, $$H$$ is characteristic in $$K$$.
 * $$H$$ is characteristic in any subgroup of $$G$$ contained in its normalizer $$N_G(H)$$.

Formalisms
A subgroup $$H \le G$$ is intermediately normal-to-characteristic in $$G$$ if and only if $$H$$ is an intermediately characteristic subgroup in $$N_G(H)$$.

Stronger properties

 * Weaker than::Intermediately automorph-conjugate subgroup
 * Weaker than::Sylow subgroup
 * Weaker than::Hall subgroup
 * Weaker than::Join of intermediately automorph-conjugate subgroups

Weaker properties

 * Stronger than::Intermediately subnormal-to-normal subgroup