A group is said to be acyclic if its reduced homology vanishes. Many interesting classes of groups have been discovered having this property ([2] is a useful survey). This is an indication that the reduced homology carries limited information. Here we obtain information about H̃(G) for a class of groups G that includes all locally finite groups and all soluble groups of finite rank. We show that...

Let C be a class of groups, closed under taking subgroups and quotients. We prove that if all metabelian groups of C are torsion-by-nilpotent, then all soluble groups of C are torsion-by-nilpotent. From that, we deduce the following consequence, similar to a well-known result of P. Hall: if H is a normal subgroup of a group G such that H and G/H ′ are (locally finite)-by-nilpotent, then G is (l...