In profinite groups, a countable cover of coprime commutators by procyclic subgroups implies the pronilpotent residual is finite-by-procyclic.
Shumyatsky, Commutators of elements of coprime orders in finite group, Forum Math.27(2015), 575–583
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Coprime commutators in profinite groups
In profinite groups, a countable cover of coprime commutators by procyclic subgroups implies the pronilpotent residual is finite-by-procyclic.