The class of Butler groups is defined to be the class of pure subgroups of finite rank completely decomposable torsion-free abelian groups. Let ${\cal K}$(0) denote the class of Butler groups. For $n\ge 1$, define the class ${\cal K}(n)$ to be those groups A which appear as the kernel of a balanced exact sequence $0\to A\to B\to C\to 0$ with B a finite rank completely decomposable group and C a ${\cal K}(n - 1)$ group. Dually, let co-${\cal K}(0)$ denote the class of Butler groups, and for $n\ge 1$ define the class co-${\cal K}(n)$ to be those groups C which appear as the quotient in a cobalanced exact sequence $0\to A\to B\to C\to 0$ with B a finite rank completely decomposable group and A a co-${\cal K}(n - 1)$ group. The classes ${\cal K}(n), n\ge 0$, and the classes co-${\cal K}(n),\ n\ge 0$, form two strictly decreasing chains of Butler groups, each of which intersects to the class of finite rank completely decomposable torsion-free abelian groups. Characterizations of ${\cal K}(n)$ groups and co-${\cal K}(n)$ groups are given in terms of direct sum decompositions of certain pure subgroups and factor groups, respectively. In addition it is shown that for any positive integers n and m the intersection of the classes ${\cal K}(n)$ and co-${\cal K}(m)$ contains a group which is neither in ${\cal K}(n + 1)$ nor in co-${\cal K}(m + 1).$
My Research
