Divisibility-closed implies powering-invariant
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 is a subgroup and is a divisibility-closed subgroup of , i.e., if is a natural number such that every element of has an root in , then every element of has a root in . Then, is also a powering-invariant subgroup of , i.e., if is a prime number such that every element of has a unique root, then every element of has a unique root in .
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.