Hereditarily 2-subnormal subgroup

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed a hereditarily 2-subnormal subgroup if, for any subgroup $$K$$ of $$H$$, $$K$$ is a defining ingredient::2-subnormal subgroup of $$G$$.

Stronger properties

 * Weaker than::Central subgroup
 * Weaker than::Abelian normal subgroup
 * Weaker than::Subgroup of abelian normal subgroup
 * Weaker than::Dedekind normal subgroup
 * Weaker than::Subgroup of Dedekind normal subgroup
 * Weaker than::Subgroup contained in the Baer norm

Weaker properties

 * Stronger than::Right-transitively 2-subnormal subgroup
 * Stronger than::Hereditarily subnormal subgroup

Facts

 * Centralizer of commutator subgroup is hereditarily 2-subnormal