Commutators in residually finite groups. (Q532606)

In a previous paper [Proc. Am. Math. Soc. 127, No. 9, 2583-2586 (1999; Zbl 0956.20017)] it has been shown that in a residually finite group \(G\) satisfying a law \([x,y]^n=1\) (\(n\) a prime power) the derived subgroup \(G'\) is locally finite. It seems difficult to prove the analogous result for arbitrary \(n\). However, this can be achieved under a somewhat stronger hypothesis. The paper contains a proof of the following result: Let \(n\) be a positive integer and \(G\) a residually finite group in which every product of at most 68 commutators has order dividing \(n\). Then \(G'\) is locally finite.