Balanced and cobalanced Butler groups (Q5936914)

Let \(\mathcal K(0)\) denote the class of Butler groups, i.e. pure subgroups of finite rank completely decomposable torsion-free Abelian groups. For \(n\geq 1\), \(\mathcal K(n)\) is the class of groups that appear as the group \(A\) in a balanced exact sequence \(E\): \(0\to A\to B\to C\to 0\) in which \(B\) is a finite rank completely decomposable group and \(C\) is a \(\mathcal K(n-1)\)-group. Dually, let co-\(\mathcal K(0)\) denote the class of Butler groups. For \(n\geq 1\), co-\(\mathcal K(n)\) is the class of Butler groups that appears as the quotient \(C\) in a cobalanced exact sequence \(E\) in which \(B\) is a finite rank completely decomposable group and \(A\) is a co-\(\mathcal K(n-1)\)-group. In sections 3 and 4 the author gives some direct sum characterizations of \(\mathcal K(n)\)-groups and co-\(\mathcal K(n)\)-groups (Theorems 3.2 and 4.6). These results generalize naturally some well-known properties of Butler groups.