Powering-invariance does not satisfy intermediate subgroup condition

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This article gives the statement, and possibly proof, of a subgroup property (i.e., powering-invariant subgroup) not satisfying a subgroup metaproperty (i.e., intermediate subgroup condition).
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Statement

It is possible to have groups H \le K \le G such that H is a powering-invariant subgroup of G but not of K.

Proof

Take:

  • G to be the group \mathbb{Q} \oplus \mathbb{Z}.
  • K to be the subgroup \mathbb{Q} \oplus 0.
  • H to be the subgroup \mathbb{Z} \oplus 0.

Then:

  • H is powering-invariant in G: G is not powered over any primes, so H is by definition powering-invariant in G.
  • H is not powering-invariant in K: K is powered over all primes, and H is not powered over any, so H is not powering-invariant in K.