Commutators and commutator subgroups in profinite groups
  Let $G$ be a profinite group. We prove that the commutator subgroup $G'$ is finite-by-procyclic if and only if the set of all commutators of $G$ is contained in a union of countably many procyclic subgroups. 
Cristina Acciarri and Pavel Shumyatsky
math.GR