Infinite Abelian Groups

by Irving Kaplansky

In the Introduction to this concise monograph, the author states his two main goals: 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." Suitable for advanced undergraduates and graduate students in mathematics, the text requires no extensive background beyond the rudiments of group theory.
Starting with examples of abelian groups, the treatment explores torsion groups, Zorn's lemma, divisible groups, pure subgroups, groups of bounded order, and direct sums of cyclic groups. Subsequent chapters examine Ulm's theorem, modules and linear transformations, Banach spaces, valuation rings, torsion-free and complete modules, algebraic compactness, characteristic submodules, and the ring of endomorphisms. Many exercises appear throughout the book, along with a guide to the literature and a detailed bibliography.

113 pages, Kindle Edition

Published December 19, 2018

Reviews

[Ron Me]

This is a math book, and this is its first review. I'm familiar with modern algebra, transfinite ordinals and Zorn's Lemma (equivalent to the Axiom of Choice), all used in the first few sections---reader beware! I had hoped that, since groups are so useful in physics (think of U1), infinite ones might be useful too. Finite Abelian groups are fairly straightforward, and it turns out that so are countably infinite ones, but the uncountably infinite ones are a real headache. And no hints of any way they could be useful, this book is strictly math. After wading my way through about half of the book, I gave up. Not for the faint of heart....