Powering-invariance 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., 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

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.