On groups with periodic products of commutators

To generalize a result of Deryabina and Kozhevnikov, we prove that for every m >= 1 there exists a group G that satisfies the identity ([x(1), y(1)]... [x(m), y(m)])(n) = 1, where n >= 2(53)m is either odd or divisible by 2(9), and whose commutator subgroup G' contains elements of infinite order.