Abelian groups. Structures and classifications (Q2417198)

Throughout this review, I will use the word `group' to denote abelian group. Although examples of finitely generated groups appear in the works of Galois, Cauchy and Abel on the theory of equations, and Gauss used their decomposition as direct sums of cyclic groups in his work on quadratic forms and number theory, it is fair to say that the study of groups as structures in their own right begins in the 1920s with the analysis of countable torsion groups by Prüfer. In the 1930s, Ulm discovered invariants of $p$-groups which bear his name. The Ulm invariant of a $p$-group $G$ is a function from ordinals to cardinals defined by $u(\kappa)=$ the dimension of the $\mathbb Z(p)$-vector space $ p^\kappa G/p^{\kappa+1}G$. Here, $p^\kappa G$ is defined by induction as $p(p^{\kappa-1}G)$ if $\kappa $ is a successor ordinal and $\bigcap_{\alpha<\kappa}p^\alpha G$ if $\kappa$ is a limit ordinal. Ulm invariants are eventually constsnt, and Ulm showed that these invariants classify those $p$-groups $G$ for which $p^\omega G=p^{\omega+1} G$, while Zippin showed exactly which functions arise as the Ulm invariants of countable $p$-groups. Also in the 1930s, Baer classified completely decomposable groups, that is, direct sums of subgroups of the rationals, but the structure of torsion-free groups in general was and remains a mystery. The case of mixed groups, i.e., those which have elements of both non-zero finite and infinite order, is equally mysterious. A major advance in the theory was the publication in 1958 of the first monograph devoted to the study of infinite groups, Laszlo Fuchs' \textit{Abelian groups}. This volume contained virtually everything that was known at that time about the structure of groups. More importantly, it contained a list of 86 open problems, which spurred a revival of interest in the topic and led to a host of theses and other publications. Even earlier, in 1954, Irving Kaplansky published his University of Michigan lecture notes under the title \textit{Infinite abelian groups}, which, while introducing several novel approaches, was not intended to be as complete as Fuchs' work. The floodgates were now open with over 300 new publications in the 60s and 70s on the structure of groups; in particular, on extensions of Ulm's and Zippin's theorems to ever larger classes of $p$-groups. It turned out that several classes of groups, described by different group-theoretic or homological methods, that were classified by Ulm invariants, were subsumed into the class of simply-presented $p$-groups, that is, those generated by an arbitrarily large class with all relations of the form $px=0$ or $px=y$. When Fuchs in 1970 and 1973 published a new edition of his work, now entitled \textit{Infinite abelian groups}, it had expanded to two volumes. Once again, the coverage was encyclopaedic; those Problems of the first volume which had been satisfactorily dealt with were omitted, but the open problems now amounted to 100. In 2015, a valiant effort of Fuchs produced a third edition, now back with its original title and published in the Springer Monographs in Mathematics Series [\textit{L. Fuchs}, Abelian groups. Cham: Springer (2015; Zbl 1416.20001)]. At 750 pages, it is a massive work, still with open problems, but no longer aiming to be a complete exposition of the literature. A major advance that is included that the Ulm invariants for all primes $p$, together with some new cardinal invariants due to Warfield, provide a classification for the class of mixed groups all of whose summands are an extension of a finite rank completely decomposable group by a simply presented group. These facts explain why the publication of a new monograph with the title \textit{Abelian groups. Structure and classification} is an audacious enterprise, which is carried out with flair in the publication under review. After a short section introducing the notation and basic properties of groups and the tools needed to study them, the first half of the book deals with the results described above of the classification of simply presented groups and certain mixed groups. The authors show that, with a generally accepted definition of classification, no larger class of groups can be so classified. Up to this point, the book covers much the same ground as Fuchs' 2015 edition. The second half is devoted mainly to developments due to the authors. While the structures discussed above are all in the context of first-order predicate logic, the authors now introduce infinitary logic, which allows sentences to contain infinitely many conjunctions and disjunctions. Since infinitary logic may be unfamiliar to algebraists, they describe its syntax and model theory before presenting its application to the classification of groups. A major feature is that we are allowed to use partial isomorphisms, and consequently classifiy groups up to partial isomorphism. The last quarter of the book is devoted to topological group theory, and in particular, topological versions of Ulm and Warfield invariants and simple presentations. This chapter includes the classification of (locally) compact groups. To summarise, this monograph is a worthy contribution to the literature on abelian groups. Each chapter contains exercises and each section historical remarks and pointers to relevant publications. While not as encyclopaedic as Fuchs' 2015 edition, the book's well defined structure makes it more suitable for study by advanced students.