An Introduction to Group Rings

1 Groups.- 1.1 Basic Concepts.- 1.2 Homomorphisms and Factor Groups.- 1.3 Abelian Groups.- 1.4 Group Actions, p-groups and Sylow Subgroups.- 1.5 Solvable and Nilpotent Groups.- 1.6 FC Groups.- 1.7 Free Groups and Free Products.- 1.8 Hamiltonian Groups.- 1.9 The Hirsch Number.- 2 Rings, Modules and Algebras.- 2.1 Rings and Ideals.- 2.2 Modules and Algebras.- 2.3 Free Modules and Direct Sums.- 2.4 Finiteness Conditions.- 2.5 Semisimplicity.- 2.6 The Wedderburn-Artin Theorem.- 2.7 The Jacobson Radical.- 2.8 Rings of Algebraic Integers.- 2.9 Orders.- 2.10 Tensor Products.- 3 Group Rings.- 3.1 A Brief History.- 3.2 Basic Facts.- 3.3 Augmentation Ideals.- 3.4 Semisimplicity.- 3.5 Abelian Group Algebras.- 3.6 Some Commutative Subalgebras.- 4 A Glance at Group Representations.- 4.1 Definition and Examples.- 4.2 Representations and Modules.- 5 Group Characters.- 5.1 Basic Facts.- 5.2 Characters and Isomorphism Questions.- 6 Ideals in Group Rings.- 6.1 Ring Theoretic Formulas.- 6.2 Nilpotent Ideals.- 6.3 Nilpotent Augmentation Ideals.- 6.4 Semiprime Group Rings.- 6.5 Prime Group Rings.- 6.6 Chain Conditions in KG.- 7 Algebraic Elements.- 7.1 Introduction.- 7.2 Idempotent Elements.- 7.3 Torsion Units.- 7.4 Nilpotent Elements.- 8 Units of Group Rings.- 8.1 Introduction.- 8.2 Trivial Units.- 8.3 Finite Groups.- 8.4 Units of ZS3.- 8.5 Infinite Groups.- 8.6 Finite Generation of U(ZG).- 8.7 Central Units.- 9 The Isomorphism Problem.- 9.1 Introduction.- 9.2 The Normal Subgroup Correspondence.- 9.3 Metabelian Groups.- 9.4 Circle Groups.- 9.5 Further Results.- 9.6 The Modular Isomorphism Problem.- 10 Free Groups of Units.- 10.1 Free Groups.- 10.2 Free Groups of Units.- 10.3 Explicit Free Groups.- 10.4 Explicit Free Groups in H.- 11 Properties of the Unit Group.- 11.1 Integral Group Rings.- 11.2 Group Algebras.