Group Ring Groups by Eric Jespers, Ángel del Río -- Read -- Imperial Library of Trantor

Index

Cover Title Copyright Preface Contents 1 Units in group rings: an introduction

2 Representations of algebras

2.1 Semisimple algebras 2.2 Splitting fields 2.3 Characteristic polynomial, trace and norm 2.4 Brauer group 2.5 Cohomology 2.6 Crossed products

3 Wedderburn decomposition of semisimple group algebras

3.1 Representations and characters of finite groups 3.2 Some operations with characters 3.3 Wedderburn components from character tables 3.4 Wedderburn components from monomial characters 3.5 Strongly monomial characters 3.6 Induction theorems 3.7 Brauer-Witt Theorem 3.8 Examples

4 Dedekind domains, valuations and orders

4.1 Localization and algebraic integers 4.2 Dedekind domains 4.3 Finitely generated modules over Dedekind domains 4.4 Extensions of Dedekind domains 4.5 Valuations 4.6 Orders 4.7 The discriminant 4.8 Brauer group of a number field

5 Thegroupofunitsofanorder

5.1 Lattices in real vector spaces 5.2 Hey’s Theorem and Dirichlet’s Unit Theorem 5.3 The group of units of an order is finitely generated 5.4 The group of units of an order is finitely presented 5.5 Subgroups of finite index

6 Cyclotomic integers

6.1 Cyclotomic fields 6.2 Cyclotomic units

7 Central units

7.1 Thegroupofcentralunitsofanorder 7.2 Large subgroups of central units: an algorithm 7.3 Bass units as generators of large groups of units

8 Generic units

8.1 Shifted cyclotomic polynomials 8.2 The group of generic units 8.3 A logarithm function 8.4 A basis of generic units for a subgroup of finite index in 8.5 Polynomials of small degree defining units

9 K-theory

9.1 Grothendieck group 9.2 The Whitehead group 9.3 Stable range condition 9.4 Whitehead group and the stable range condition 9.5 Applications of K-theory to units

10 General linear groups of degree 2

10.1 Number theoretical results 10.2 Normality of in 10.3 The factor group by 10.4 The group E2 (I) is of finite index in SL2 (R)

11 Generators of unit groups of group rings

11.1 Bass Unit Theorem 11.2 Generalized bicyclic units and Bass units I 11.3 Bicyclic units and Bass units 11.4 Fixed point free groups and Frobenius complements 11.5 Group rings of nilpotent groups

12 Exceptional simple components

12.1 Components of index one 12.2 Components of index two 12.3 Generalized bicyclic units and Bass units II 12.4 Normal closure of the trivial units 12.5 Normal complements 12.6 Examples: metacyclic groups 12.7 Examples with insufficient Bass units and bicyclic units

13 Idempotents and central units in group rings

References Index of Notation Index