Locally (soluble-by-finite) groups with all proper non-nilpotent subgroups of finite rank Original Research Article

A group G is said to have finite rank r if every finitely generated subgroup of G is at most r-generator. If c is a positive integer we let image denote the class of nilpotent groups of class at most c, and image the class of groups in which every proper non-image subgroup has finite rank. Our main theorem shows that if G is a locally (soluble-by-finite) group in the class image then either G is nilpotent of class at most c or G has finite rank. An analogous result holds for locally soluble (image2)*-groups, where image2 denotes the class of metabelian groups. We give an example to show that locally finite (image2)*-groups need neither have finite rank nor be metabelian.