The Fundamental Theorem of Finite Abelian Groups tells us that every finite abelian group is the direct product of cyclic groups. In most undergraduate (and even graduate) algebra classes, we end our discussion of finitely generated abelian groups there, to move on to nonabelian groups or to leave the study of groups altogether. But, according to Sàndor Szabò's new book Topics in Factorization of Abelian Groups, the discussion should not end here as there is much more to be said about how these groups factor, from both a theoretical and a computational standpoint.
The first chapter of his book gives a wonderful introduction to the topic and a survey of the types of questions that one can ask. In particular, he discusses a theorem of Hajós from 1941 which is a group theoretical equivalent of a geometric conjecture of Minkowski; it says that if a finite abelian group G is a direct product of cyclic subsets then one or more of these subsets is actually a subgroup of G. Szabó discusses several generalizations of this theorem, most of which replace the cyclic condition by various weaker conditions. The most important of these generalizations is due to Réidi and says that if G can be written as a direct product of subsets each of which contains the identity and is of prime order then (at least) one of these subsets must be a subgroup of G.
Most of the book under review deals with the proofs and generalizations of the theorems of Hajós and Réidi, and with the technical machinery needed to study them. Unfortunately, the level of exposition in the rest of the book is not as high as that of the first section. Some of this is due to problems with the editing of the book: there are a number of typos, and many more awkward grammatical constructions. Moreover, there are many places where things could be explained better than they are, and I would get lost due to some poor choices of notation or skipped steps. But my biggest complaint is that many of the topics seem contrived and disconnected, and there are a number of places where the author introduces technical machinery before explaining why it will be useful to prove results. Szabó also does not do a good job of explaining to the reader why they should care about these topics. He often alludes to applications in coding theory and number theory, but most of these are only touched on briefly.
There are enough glimmers of exciting and deep topics in this book to convince me that the author is correct that there is interesting mathematics and exciting questions to ask about finite abelian groups — and any student taking group theory would do well to read the opening section — but unfortunately this book does not do as good a job of answering many of those questions, or of explaining well those questions which it does answer.
Darren Glass (email@example.com) is an Assistant Professor at Gettysburg College. His mathematical interests include number theory, cryptography, algebraic geometry, and Galois theory.