Powering-invariance is not commutator-closed

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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., commutator-closed subgroup property).
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Statement

It is possible to have a group G and subgroups H,K of G such that both H and K are powering-invariant subgroups of G but the commutator [H,K] is not powering-invariant.

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).
  • [H,K] is an infinite cyclic group, isomorphic to the group of integers. It is not powered over any prime, hence is not powering-invariant in G.