Divisibility-closedness is strongly join-closed in nilpotent group

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 (?))
View all subgroup metaproperty satisfactions in nilpotent groups

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View all subgroup metaproperty satisfactions in nilpotent groups

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View all subgroup metaproperty satisfactions

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View all subgroup metaproperty dissatisfactions

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 $⟨H_i{⟩}_{i\in I}$ is also a divisibility-closed subgroup of $G$.

Related facts

• Powering-invariance is strongly join-closed in nilpotent group
• Divisibility-closedness is not finite-join-closed (the examples for this are solvable, but cannot be nilpotent)
• Divisibility-closedness is not finite-intersection-closed (there is an abelian example)
• Powering-invariance is strongly intersection-closed

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.