Asymptotically fixed-depth join-transitively subnormal subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed an asymptotically fixed-depth join-transitively subnormal subgroup if there exists a natural number $$n$$ such that for any $$k \ge n$$, and any $$k$$-subnormal subgroup $$K$$ of $$G$$, the join $$\langle H, K \rangle$$ is also a $$k$$-subnormal subgroup of $$G$$.

Stronger properties

 * Weaker than::Normal subgroup
 * Weaker than::Subgroup of nilpotent group
 * Weaker than::Subnormal subgroup of finite index: Also related:
 * Weaker than::Subnormal subgroup of finite group

Weaker properties

 * Stronger than::Linear-bound join-transitively subnormal subgroup
 * Stronger than::Polynomial-bound join-transitively subnormal subgroup
 * Stronger than::Join-transitively subnormal subgroup
 * Stronger than::Subnormal subgroup