Nilpotent subgroups and the hypercentre of infinite soluble groups (Q1119739)

This article answers questions posed by the second author [in Lect. Notes Math. 1281, 103-106 (1987; Zbl 0629.20014)] by constructing a finitely generated abelian-by-nilpotent-by-abelian group G with non-nilpotent hypercentre, so that every nilpotent subgroup of G is contained in a strictly larger nilpotent subgroup of G. In addition, every finitely generated non-nilpotent subgroup of G has non-nilpotent hypercentre. By contrast, finitely generated nilpotent-by-nilpotent groups have nilpotent hypercentres, but it is possible that there exist nilpotent-by-abelian- by-nilpotent groups in which every nilpotent subgroup is contained in a strictly larger nilpotent subgroup.