Group Ring Groups: Volume 1 Orders and Generic Constructions of Units

Description

This two-volume graduate textbook gives a comprehensive, state-of-the-art account of describing large subgroups of the unit group of the integral group ring of a finite group and, more generally, of the unit group of an order in a finite dimensional semisimple rational algebra. Since the book is addressed to graduate students as well as young researchers, all required background on these diverse areas, both old and new, is included. Supporting problems illustrate the results and complete some of the proofs. Volume 1 contains all the details on describing generic constructions of units and the subgroup they generate. Volume 2 mainly is about structure theorems and geometric methods. Without being encyclopaedic, all main results and techniques used to achieve these results are included. Basic courses in group theory, ring theory and field theory are assumed as background. Elementary techniques are introduced to built up an intuition for the topic Gives an overview and structure to a vibrant field of mathematics Preface Contents 1. Units in group rings: an introduction 2. Representations of algebras 3. Wedderburn decomposition of semisimple group algebras 4. Dedekind domains, valuations and orders 5. The group of units of an order 6. Cyclotomic integers 7. Central units 8. Generic units 9. K-theory 10. General linear groups of degree 2 11. Generators of the unit group of an integral group ring 12. Exceptional simple components 13. Idempotents and central units in group rings References Index of Notation Index