Divisibility-closed implies powering-invariant

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This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., divisibility-closed subgroup) must also satisfy the second subgroup property (i.e., powering-invariant subgroup)
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Suppose G is a subgroup and H is a divisibility-closed subgroup of G, i.e., if n is a natural number such that every element of G has an n^{th} root in G, then every element of H has a n^{th} root in H. Then, H is also a powering-invariant subgroup of G, i.e., if p is a prime number such that every element of G has a unique p^{th} root, then every element of H has a unique p^{th} root in H.

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This statement is pretty direct. Note that existence follows from being divisibility-closed, and uniqueness in the whole group implies uniqueness in the subgroup as well.