Bulletin of the Australian Mathematical Society

Table of Contents - August 2009 - Volume 80, Issue 01  

Research Article

DIAGRAMS OF AN ABELIAN GROUP

THEODORE G. FATICONIa1

a1 Department of Mathematics, Fordham University, Bronx, NY 10458, USA (email: faticoni@fordham.edu)

Abstract

In this paper, we characterize quadratic number fields possessing unique factorization in terms of the power cancellation property of torsion-free rank-two abelian groups, in terms of Σ-unique decomposition, in terms of a pair of point set topological properties of Eilenberg–Mac Lane spaces, and in terms of the sequence of rational primes. We give a complete set of topological invariants of abelian groups, we characterize those abelian groups that have the power cancellation property in the category of abelian groups, and we characterize those abelian groups that have Σ-unique decomposition. Our methods can be used to characterize any direct sum decomposition property of an abelian group.

(Received June 16 2008)

2000 Mathematics subject classification

  • primary 20K99; secondary 16K99

Keywords and phrases

  • abelian groups;
  • point set topological spaces;
  • homology;
  • homotopy;
  • rings and modules;
  • categories;
  • functors