Right-transitively fixed-depth subnormal subgroup

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed right-transitively fixed-depth subnormal if there exists $$k \ge 1$$ such that $$H$$ is right-transitively $$k$$-subnormal in $$G$$: whenever $$K$$ is a $$k$$-subnormal subgroup of $$H$$, $$K$$ is also $$k$$-subnormal in $$G$$.

Any right-transitively $$k$$-subnormal subgroup is also right-transitively $$l$$-subnormal for $$l \ge k$$.

(Here, a $$k$$-subnormal subgroup is a defining ingredient::subnormal subgroup whose defining ingredient::subnormal depth is at most $$k$$).

Stronger properties

 * Weaker than::Transitively normal subgroup: Here, $$k = 1$$. Also related:
 * Weaker than::Central subgroup
 * Weaker than::Central factor
 * Weaker than::Direct factor
 * Weaker than::Conjugacy-closed normal subgroup
 * Weaker than::SCAB-subgroup
 * Weaker than::Right-transitively 2-subnormal subgroup: Here, $$k = 2$$. Also related:
 * Weaker than::Subgroup of abelian normal subgroup
 * Weaker than::Abelian normal subgroup
 * Weaker than::Base of a wreath product
 * Weaker than::Normal T-subgroup
 * Weaker than::Nilpotent subnormal subgroup
 * Weaker than::Finite subnormal subgroup

Weaker properties

 * Stronger than::Subnormal subgroup

Related properties

 * Left-transitively fixed-depth subnormal subgroup