Strong conciseness of coprime and anti-coprime commutators

A coprime commutator in a profinite group G is an element of the form [x, y], where x and y have coprime order and an anti-coprime commutator is a commutator [x, y] such that the orders of x and y are divisible by the same primes. In the present paper, we establish that a profinite group G is finite-by-pronilpotent if the cardinality of the set of coprime commutators in G is less than 2ℵ0. Moreover, a profinite group G has finite commutator subgroup G′ if the cardinality of the set of anti-coprime commutators in G is less than 2ℵ0.