Characteristic subgroup of abelian group not implies local powering-invariant

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This article gives the statement and possibly, proof, of a non-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) need not satisfy the second subgroup property (i.e., local powering-invariant subgroup)
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This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties, when the big group is a abelian group. That is, it states that in a abelian group, every subgroup satisfying the first subgroup property (i.e., characteristic subgroup) need not satisfy the second subgroup property (i.e., local powering-invariant subgroup)
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Statement

It is possible to have an abelian group G and a characteristic subgroup H of G that is not a local powering-invariant subgroup of G: in other words, it is possible to have h \in H and n \in \mathbb{N} are such that there is a unique x \in G satisfying x^n = h, but x \notin H.

Related facts

Similar facts

Opposite facts

Proof

Set G = \mathbb{Z} and H as the subgroup 2\mathbb{Z}.

  • H is charcateristic in G: In fact, H is a verbal subgroup of G.
  • H is not local powering-invariant in G: The element 2 \in H has a unique square root (corresponding to n = 2) in G, but this square root is not in H.