Endomorphism image 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., endomorphism image) must also satisfy the second subgroup property (i.e., powering-invariant subgroup)
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Suppose G is a group and H is a subgroup of G that is an endomorphism image of G, i.e., there exists an endomorphism \sigma of G such that \sigma(G) = H. Then, H is a powering-invariant subgroup of G, i.e., if G is powered over a prime p (i.e., every element of G has a unique p^{th} root, so is H.

Facts used

  1. Endomorphism image implies divisibility-closed
  2. Divisibility-closed implies powering-invariant


The proof follows directly from Facts (1) and (2).