On finitely presented soluble groups with small abelian‐by‐finite images

The main result of [7] asserted that if G is a finitely presented soluble group and no subgroup of finite index maps epimorphically to Z then G is virtually nilpotent-by-cyclic. One important ingredient in the proof was Theorem A of Brookes [2]. We have recently become aware that Brookes’ result requires stronger hypotheses than those stated in [2]. As a consequence, the proof of the above assertion is lacunary, and indeed counter-examples have been found by Victor Bludov. Let p be a prime and let G be the group with presentation