Let C C be a class of groups, closed under taking subgroups and quotients. We prove that if all metabelian groups of C C are torsion-by-nilpotent, then all soluble groups of C C are torsion-by-nilpotent. From that, we deduce the following conse-Ž quence, similar to a well-known result of P. Hall 1958, Illinois J. Math. 2, . 787᎐801 : if H is a normal subgroup of a group G such that H and GrHЈ are Ž . Ž . locally finite -by-nilpotent, then G is locally finite -by-nilpotent. We give an Ž . example showing that this last statement is false when '' locally finite -by-nilpotent'' is replaced with ''torsion-by-nilpotent.'' ᮊ 2001 Academic Press THEOREM A. Let H be a normal subgroup of a group G. If GrHЈ and H are nilpotent, then G is nilpotent.
This result is often very useful for proving that a group is nilpotent. In particular, by an induction on the derived length, it is easy to obtain the following consequence. THEOREM B. Let C C be a class of groups which is closed under taking subgroups and quotients. Suppose that all metabelian groups of C C are nilpotent. Then all soluble groups of C C are nilpotent. 1 The second author thanks the European Community for their support with a Marie Curie Grant.