Characteristic subgroup of abelian group 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., characteristic subgroup of abelian group) must also satisfy the second subgroup property (i.e., powering-invariant subgroup)
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Statement

Suppose G is an abelian group and H is a characteristic subgroup of G. Then, H is a powering-invariant subgroup of G: for any prime number p such that every element of G has a unique p^{th} root, every element of H also has a unique p^{th} root in H.

Related facts

Facts used

  1. Abelian implies universal power map is endomorphism

Proof

Proof idea

The idea is to use Fact (1), and the powering, to show that the p^{th} power map is an automorphism, hence so is its inverse (the p^{th} root map), and hence, because the subgroup is characteristic, it is invariant under the map.