Kaplansky: Infinite abelian groups Introduction
In the early days of group theory attention was confined almost entirely to finite groups. But recently, and above all in the last two decades, the infinite group has come into its own. The results obtained on infinite abelian groups have been particularly penetrating. This monograph has been written with two objectives in mind: first, to make the theory of infinite abelian groups available in a convenient form to the mathematical public; second, to help students acquire some of the techniques used in modern infinite algebra.
For this second purpose infinite abelian groups serve admirably. No extensive background is required for their study, the rudiments of group theory being sufficient. There is a good variety in the transfinite tools employed, with Zorn's lemma being applied in several different ways. The traditional style of transfinite induction is not completely ignored either, for there is a theorem whose very formulation uses transfinite ordinals. The peculiar role sometimes played by a countability hypothesis makes a challenging appearance.
It is furthermore helpful that finite abelian groups are completely known. In other subjects, such as rings or non-abelian groups, there are distracting difficulties which occur even in the finite case. Here, however, our attention is concentrated on the problems arising from the fact that the groups may be infinite.
With a student audience in mind, I have given details and included remarks that would ordinarily be suppressed in print. However, as the discussion proceeds it becomes somewhat more concise. A serious effort has been made to furnish, in brief space, a reasonably complete account of the subject. In order to do this, I have relegated many results of some interest to the role of exercises, and a large part of the literature is merely surveyed in the guide to it provided in section 6 20.
This material is adapted from a course which I gave at the University of Chicago in the fall of 1950. I should like to record my indebtedness to the many able members of that class, particularly to George Backus, Arlen Brown, and Roger Farrell. Thanks are expressed to Isidore Fleischer for the ideas in section 6 16 (the torsion-free case of Theorem 22 was discovered by him and appears in his doctoral dissertation); to Robert Heyneman and George Kolettis, who read a preliminary version of this work and made many valuable suggestions; to Tulane University and the University of Michigan, where I had the opportunity to lecture on abelian groups; and to the Office of Naval Research.
A special acknowledgment goes to Professor Reinhold Baer. It was from his papers that I learned much of the theory of abelian groups. Furthermore, when this monograph was nearly complete, I had the privilege of reading an unpublished manuscript (of book length) on abelian groups which he prepared several years ago.
JOC/EFR March 2006
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