Divisibility-closedness is strongly join-closed in nilpotent group

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This article gives the statement and possibly, proof, of a subgroup property satisfying a subgroup metaproperty, when the big group is a nilpotent group. That is, it states that in a Nilpotent group (?), the subgroup property (i.e., Divisibility-closed subgroup (?)) satisfies the metaproperty (i.e., Strongly join-closed subgroup property (?))
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Statement

Suppose G is a nilpotent group and H_i, i \in I are all divisibility-closed subgroups of G. Then, the join of subgroups \langle H_i \rangle_{i \in I} is also a divisibility-closed subgroup of G.

Related facts

Facts used

  1. Divisible subset generates divisible subgroup in nilpotent group

Proof

The proof follows from Fact (1): simply take the set-theoretic union of the subgroups as the "divisible subset" for the appropriate set of primes and argue that the subgroup generated by it is also appropriately divisible.