Milne J. Group Theory 2025

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Milne J. Group Theory 2025

Textbook in PDF format November 6, 2025. This book by J.S. Milne provides a comprehensive course in group theory, beginning with the absolute fundamentals of definitions, examples, and basic results like homomorphisms, cosets, and normal subgroups. It then progresses systematically into more advanced topics, starting with the construction of groups via free groups and presentations, including a study of Coxeter groups. The text continues by exploring the internal structure of groups through automorphisms, characteristic subgroups, and how groups can be built from others using semidirect products and extensions. A significant focus is placed on groups acting on sets, which leads into the powerful Sylow theorems and their critical applications for understanding finite groups. The theory of group structure is further developed with an in-depth look at subnormal series and the important classes of solvable and nilpotent groups. The book culminates in a thorough introduction to the representation theory of finite groups, covering key results like Maschke's Theorem and developing the theory of characters and character tables. Each chapter is supported by a set of exercises, with solutions provided at the end, making it a self-contained resource for learning the subject. These notes give a exposition of the theory of groups, including free groups and Coxeter groups, the Sylow theorems, and the representation theory of finite groups. They originated as the notes for a first-year graduate course taught at the University of Michigan, but they have since been revised and expanded numerous times. The only prerequisite is an undergraduate course in abstract algebra. There are over a hundred exercises, many with solutions. Basic Definitions and Results Definitions and examples Multiplication tables Subgroups Groups of small order Homomorphisms Cosets Normal subgroups Kernels and quotients Theorems concerning homomorphisms Direct products Commutative groups The order of ab Exercises Free Groups and Presentations; Coxeter Groups Free monoids Free groups Generators and relations Finitely presented groups Coxeter groups Exercises Automorphisms and Extensions Automorphisms of groups Characteristic subgroups Semidirect products Extensions of groups The Hölder program Exercises Groups Acting on Sets Definition and examples Permutation groups The Todd-Coxeter algorithm Primitive actions Exercises The Sylow Theorems; Applications The Sylow theorems Alternative approach to the Sylow theorems Examples Exercises Subnormal Series; Solvable and Nilpotent Groups Subnormal Series Solvable groups Nilpotent groups Groups with operators Krull-Schmidt theorem Exercises Representations of Finite Groups Matrix representations Roots of 1 in fields 96 Linear representations Maschke’s theorem The group algebra; semisimplicity Semisimple modules Simple F-algebras and their modules Semisimple F-algebras and their modules The representations of G The characters of G The character table of a group Examples Exercises Additional Exercises Solutions to the Exercises Two-Hour Examination Bibliography Index