Groups assembled from free and direct products

Let A be the collection of groups which can be assembled from infinite cyclic groups using the binary operations free and direct product. These groups can be described in several ways by graphs. The group (Z ∗Z)×(Z ∗Z) has been shown by [1] to have a rich subgroup structure. In this article we examine subgroups of A–groups. DefineA to be the smallest class of groups which contains the infinite cyclic group Z, and which contains both the free product and the direct product of any two of its members. So A contains f.g. free groups, f.g. free abelian groups, direct products of f.g. free groups, free products of f.g. free abelian groups, etc., see Figure 1. In particular, groups in A are finitely generated. Groups in A are examples of Z @@ Z = Z × · · · × Z Fn = Z ∗ · · · ∗ Z Z1 ∗ · · · ∗ Zk Fn1 × · · · × Fnk