C-closed implies completely divisibility-closed in nilpotent group

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This article gives the statement and possibly, proof, of an implication relation between two subgroup properties, when the big group is a nilpotent group. That is, it states that in a Nilpotent group (?), every subgroup satisfying the first subgroup property (i.e., C-closed subgroup (?)) must also satisfy the second subgroup property (i.e., Completely divisibility-closed subgroup (?)). In other words, every c-closed subgroup of nilpotent group is a completely divisibility-closed subgroup of nilpotent group.
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Statement

Suppose G is a nilpotent group and H is a c-closed subgroup of G, i.e., H equals the centralizer C_G(K) for some subgroup K of G. Then, H is a completely divisibility-closed subgroup of G: for any prime number p such that every element of G has a p^{th} root in G, it is true that all p^{th} roots of any element of H are in H.

Related facts

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