Introduction

In this article we give a brief survey of recent work of the authors and others. We recall that if r is a fixed positive integer then a group G has (Prüfer) rank r if every finitely generated subgroup of G can be generated by r elements and r is the least such integer. Throughout this paper we shall say that a group G has rank r if, in the above sense, it has rank at most r. We shall be primarily concerned with those groups which have the property that certain of their subgroups have finite rank, although in the last section we discuss what might be thought of as the dual problem.

The types of question we have in mind are these: suppose that X is some class of groups and let G be a group in which every proper subgroup belongs to X. What can be said about G? Is G necessarily in X? Can we classify those groups G that are not in X? What if we relax the condition on G and only suppose that certain distinguished subgroups of G belong to X?