Noetherian implies subnormal join property

This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., Noetherian group) must also satisfy the second group property (i.e., group satisfying subnormal join property)
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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. Noetherian implies ascending chain condition on subnormal subgroups
2. Ascending chain condition on subnormal subgroups implies subnormal join property

Proof

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