Groups assembled from free and direct products
Let \(A\) 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. These groups can be described in several ways by graphs. In this paper the subgroups of \(A\)-groups are examined. A finite graph is called an assembly if it belongs to the smallest collection of graphs containing the one-point graph \(K_ 1\), and which is closed under the operations ``disjoint union'' and ``join''. Let \(A\) be an assembly, and let \(\Gamma\) be a finite graph. If there is a monomorphism \(f: F_ \Gamma\to F_ A\), then \(\Gamma\) is an assembly. Let \(A\) and \(A'\) be finite assemblies and let \(f: F_{A'}\to F_ A\) be a monomorphism. Then \(A'\leq A\).
scientific article; zbMATH DE number 428127