An Introduction to Group Rings

Inhaltsverzeichnis

• 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.