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Published online by Cambridge University Press: 03 February 2005
The following theorem is proved. For any positive integers $k$ and $n$ there exists $t$ depending only on $k$ and $n$ such that the class of all groups $G$ having $\gamma_k(G)$ locally finite and satisfying the condition that the product of any $t$ commutators of the form $[x_1,x_2,{\dots}\,,x_k]$ is of order dividing $n$ is a variety.