The Structure Theorems for Infinite Abelian Groups

  • Boyd Mwalungali Department of Mathematics Copperbelt University 21692 Kitwe, Zambia.
  • William Sakala
Keywords: groups, finite and infinite abelian groups, finitely generated groups, divisible groups


This paper, examines one of the most fundamental and interesting algebraic structures, infinite abelian groups, from the perspective of group theory The Theory of abelian groups is generally simpler than that of their non-abelian counterparts and finite abelian groups are very well understood. Then would like to state whether the theory of Structure for infinite abelian group of additive rational number is finitely generated or a divisible abelian group and also show examples of each classification. Thus, classify groups by stating whether elements are finite or infinite and categories them as infinite abelian Groups. Moreover, an application to homology group and rotation in two dimensions is presented, as a demonstration of the structure of infinite abelian groups utility.


. Kaplausky, I.1968.Infinite abelian groups. University of Michigan Press, Ann Arbor, Michigan.89p.

. J.B Fraleigh, A First Course in Abstract Algebra, 7th edition, Addison Wesley, 2003

. I.N.Herstein, Topics in Algebra, 2nd edition, John Wiley& Sons 1975

. Joseph J. Rotman, An Introduction to the Theory of Groups. 4th Edition, New York, New York, Springer-Verlag, 1995.

. Bing, some aspects of the topology of 3-manifolds related to the Poincare conjecture. In lectures On modern mathematics, Vol. (ii), T.L Scaty(Ed), New York, 93-128

. Rotman, Joseph J.1965.The theory of Groups: an Introduction, Allyn and Bacon, Inc, Boston, Massachusetts. 304p

. L. Fuchs, "Infinite abelian groups”, 2, Acad. Press (1973)

. Thomas W. Hungerford, Algebra. New York, New York Holt, Rinehart and Winston, 1974.

. David S. Dummit, Richard M. Foote, Abstract Algebra. 2nd Edition, New York, New York, John Wiley and Sons, 1999.

. Bredon, G. (1993). Topology & Geometry. New York: Springer-Verlag New York, Inc.

. W. Keith Nicholson, Introduction to Abstract Algebra. 2nd Edition New York, New York, John Wiley and Sons, 1999.

. P.Garret,Website:,last modified 2013.