Date of Award
Master of Science (MS)
Mathematics and Statistics
E. E. Underwood
When the theory of groups was first introduced, the attention was on finite groups. Now, the infinite abelian groups have some into their own. the results obtained in infinite abelian groups are very interesting and penetrating in other branches of Mathematics. For example, every theorem that is stated in this paper may be generalized for modules over principal ideal domains and applied to the study of linear transformations.
This paper presents the most important results in infinite abelian groups following the exposition given by J. Rotman in his book, Theory of Groups: An Introduction. Also, some of the exercises given by J. Rotman are presented in this paper. In order to facilitate our study, two classifications of infinite abelian groups are used. The first reduces the study of abelian groups to the study of torsion groups, torsion-free groups and an extension problem. The second classification reduces to the study of divisible and reduced groups. Following this is a study of free abelian groups that are, in a certain sense, dual to the divisible groups; the basis and fundamental theorems of finitely generated abelian groups are proved. Finally, torsion groups and torsion-free groups of rank 1 are studied.
It is assumed that the reader is familiar with elementary group theory and finite abelian groups. Zorn's lemma is applied several times as well as some results of vector spaces.
Pascual, Joaquin, "Infinite Abelian Groups" (1970). All Graduate Plan B and other Reports. 1122.