Balanced Butler Groups

Abstract

The class of Butler groups, pure subgroups of finite rank completely decomposable groups, has been studied extensively by abelian group theorists in recent years. Classification by numerical invariants up to quasi-isomorphism and even isomorphism has been achieved for special subclasses. Here we highlight a new class in which to extend and expand classification results, the balanced Butler groups or K(1)-groups. These are the pure balanced subgroups of finite rank completely decomposable groups. A strictly decreasing chain of classes of Butler groups, introduced by Kravchenko, is obtained by defining the K(n)-groups (n≥2) to be those balanced subgroups of a completely decomposable group for which the quotient is a K(n−1)-group. We establish an internal characterization of K(n)-groups, give a method for constructing examples, and derive decomposition results.