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Powering-invariance is not finite-join-closed

Statement

It is possible to have a group G and subgroups H and K of G such that both H and K are both powering-invariant subgroups of G but the join of subgroups \langle H, K \rangle is not a powering-invariant subgroup of G.

Related facts

Proof

Suppose G is the generalized dihedral group corresponding to the additive group of rational numbers. Let H and K both be subgroups of order two generated by different reflections. Then, the following are true:

  • G is powered over all primes other than 2.
  • H and K are both powering-invariant subgroups on account of being finite subgroups (see finite implies powering-invariant).
  • \langle H, K \rangle is isomorphic to the infinite dihedral group. It is not powered over any primes, and in particular it is not powering-invariant in G.