Balanced and Cobalanced Butler Groups

The class of Butler groups—pure subgroups of finite-rank torsion-free completely decomposable groups—has been widely studied by abelian group theorists. Here we present several classification results for groups in two chains of special subclasses of the Butler groups, the (n)- and co- (n)-groups. The classes (n) and co- (n), n ≥ 1, are defined by balanced and cobalanced exact sequences of Butler groups, respectively. We give direct sum decompositions of certain pure subgroups of (n)-groups and certain torsion-free quotients of co- (n)-groups. The direct sum decompositions characterize the groups in those classes and extend some well-known results for Butler groups.