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Let γ i (G) and Z i (G) denote the i-th terms of the lower and upper central series of a group G, respectively. In 1956 P. Hall showed that if γ i+1 (G) is finite, then the index |G: Z 2i (G)| is finite. We prove that the same result holds under the weaker hypothesis that |γ i+1 (G): γ i+1 (G) n Z i (G)| is finite.

Generalizing a theorem of P. Hall on finite-by-nilpotent groups