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Additive group theory and non-unique factorizations. (English) Zbl 1221.20045
Geroldinger, Alfred et al., Combinatorial number theory and additive group theory. With a foreword by Javier Cilleruelo, Marc Noy and Oriol Serra (Coordinators of the DocCourse). Basel: Birkhäuser (ISBN 978-3-7643-8961-1/pbk; 978-3-7643-8962-8/ebook). Advanced Courses in Mathematics - CRM Barcelona, 1-86 (2009).

From the introduction: The course centers on the interaction between two, at first glance very disparate areas of mathematics: non-unique factorization theory and additive group theory.

The main objective of factorization theory is a systematic treatment of phenomena related to the non-uniqueness of factorizations in monoids and integral domains. In the setting of Krull monoids (the main examples we have in mind are the multiplicative monoids of rings of integers of algebraic number fields) most problems can be translated into zero-sum problems over the class group. It will be a main aim of this course to highlight this relationship.

In Section 1 we introduce the basic concepts of factorization theory, point out that arithmetical questions in arbitrary Krull monoids can be translated into combinatorial questions on zero-sum sequences over the class group and formulate a main problem. In Section 2 we introduce the Davenport constant, and using group algebras we derive its precise value for $p$-groups (Theorem 2.10). In Section 3 we discuss the structure of sets of lengths. The characterization problem is a central topic. We give a proof in the case of cyclic groups, and this proof requires most of the results from additive group theory discussed in this course. Section 4 starts with addition theorems, and then the invariants $\eta \left(G\right)$ and $s\left(G\right)$ – occurring in the Theorem of Erdős-Ginzburg-Ziv – are studied. We outline the power of the inductive method and determine the invariants $D\left(G\right)$, $\eta \left(G\right)$ and $s\left(G\right)$ for groups of rank two. Section 5 deals with inverse zero-sum problems. The focus is on cyclic groups and on groups of rank two.

MSC:
 20M14 Commutative semigroups 11B13 Additive bases, including sumsets 11B75 Combinatorial number theory 11A51 Factorization; primality 13A05 Divisibility 11R27 Units and factorization 20M13 Arithmetic theory of monoids 13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations 20M05 Free semigroups, generators and relations, word problems 20D60 Arithmetic and combinatorial problems on finite groups