Title

Characterizations of balanced and cobalanced Butler groups

Date of Completion

January 1996

Keywords

Mathematics

Degree

Ph.D.

Abstract

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-${\cal K}(m + 1).$ ^