Linear-bound join-transitively subnormal subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed linear-bound join-transitively subnormal if there exists a natural number $$n$$ such that, given any $$k$$-defining ingredient::subnormal subgroup $$K$$ of $$G$$, the join $$\langle H, K \rangle$$ is $$nk$$-subnormal.

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

Stronger properties

 * Weaker than::Normal subgroup: For a normal subgroup, we can set $$n = 1$$.
 * Weaker than::2-subnormal subgroup: For a 2-subnormal subgroup, we can set $$n = 2$$.
 * Weaker than::Subnormal subgroup of finite index
 * Weaker than::Intermediately linear-bound join-transitively subnormal subgroup
 * Weaker than::Linear-bound intermediately join-transitively subnormal subgroup
 * Weaker than::Asymptotically fixed-depth join-transitively subnormal subgroup

Weaker properties

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

Metaproperties
This follows from the fact that normal subgroups are linear-bound join-transitively subnormal, but not all subnormal subgroups are linear-bound join-transitively subnormal. That in turn follows from the fact that subnormality is not finite-join-closed.

If $$H_1, H_2 \le G$$ are both linear-bound join-transitively subnormal with corresponding natural numbers $$n_1, n_2$$, their join is linear-bound join-transitively subnormal with natural number $$n_1n_2$$. (A smaller natural number might also work for the join).