Noetherian implies subnormal join property

Statement
Any Noetherian group (also called slender group, i.e., a group in which every subgroup is finitely generated) satisfies the subnormal join property: a join of finitely many subnormal subgroups in the group is subnormal.

Facts used

 * 1) uses::Noetherian implies ascending chain condition on subnormal subgroups
 * 2) uses::Ascending chain condition on subnormal subgroups implies subnormal join property

Proof
The proof follows directly from facts (1) and (2).