Join of finitely many subnormal subgroups

Definition
A subgroup $$H$$ of a group $$G$$ is termed a join of finitely many subnormal subgroups if there exist finitely many defining ingredient::subnormal subgroups $$H_1, H_2, \dots, H_n$$ of $$G$$ such that $$H$$ is the join $$\langle H_1, H_2, \dots, H_n \rangle$$.

Note that in a group satisfying subnormal join property, being a join of finitely many subnormal subgroups is precisely equivalent to being a subnormal subgroup.

Stronger properties

 * Weaker than::Subnormal subgroup

Weaker properties

 * Stronger than::Join of subnormal subgroups

Metaproperties
Suppose $$H \le K \le G$$ are groups such that $$H$$ is the join of finitely many subnormal subgroups in $$G$$. Then, $$H$$ is the join of finitely many subnormal subgroups in $$K$$.