Powering-invariance 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., Powering-invariant 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 powering-invariant subgroups of $G$. Then, the join of subgroups $\langle H_{i}\rangle _{i\in I}$ is also a powering-invariant subgroup of $G$.

Related facts

• Powering-invariance is strongly intersection-closed
• Powering-invariance is not finite-join-closed
• Divisibility-closedness 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)

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 and hence appropriately powered.