Current Version (4.01, 2021). pdf file
Version 3.11 pdf file formatted for ereaders (9pt; 89mm x 120mm; 5mm margins)

These notes give a concise 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.

I have made Version 4.0, including the source files, available under a Creative Commons licence CC BY-NC-SA 4.0. This means that the work is not only freely available, but also freely editable. Roughly speaking, the licence allows you to edit and redistribute this text in any way you like, as long as you include an accurate statement about authorship and copyright, do not use it for commercial purposes, and distribute it under this same licence.

Contents

1. Basic Definitions and Results
2. Free Groups and Presentations; Coxeter Groups
3. Automorphisms and Extensions
4. Groups Acting on Sets
5. The Sylow Theorems; Applications
6. Subnormal Series; Solvable and Nilpotent Groups
7. Representations of Finite Groups

Prerequisites

An undergraduate "abstract algebra" course.

Translations

Arabic translation Translation by Ahmad Alkhalaf-Iman Taha and others, Al-Baath University, Syria

History

v2.01. (August 21, 1996). First version on the web; 57 pages.
v2.10. (January 28, 2002). Fixed misprints; made many improvements to the exposition; added an index, 80 exercises (30 with solutions), and an examination; 86 pages.
v2.11. (August 29, 2003). Fixed many minor errors; numbering unchanged; 85 pages.
v3.00. (September 1, 2007). Revised and expanded; 121 pages.
v3.01. (May 17, 2008). Very minor fixes and improvements; 124 pages.
v3.02. (Sept. 21, 2009). Minor corrections; changed TeX styles; 127 pages. pdf file for v3.02
v3.10. (Sept. 24, 2010). Many minor improvements; 131 pages. pdf file for v3.10
v3.11. (March 28, 2011). Minor additions; 135 pages.
v3.12. (April 9, 2012). Minor fixes; 133 pages.
v3.13. (March 15, 2013). Minor fixes; 135 pages.
v3.14. (March 17, 2017). Minor fixes; 135 pages.
v3.15. (March 20, 2020). Corrected proof of Theorem 2.16; minor fixes; 135 pages. pdf file for v3.15
v3.16. (July 16, 2020). Minor revision of Chapter 7; 137 pages. pdf file for v3.16
v4.00. (June 23, 2021). Made work, including the source files, available under a Creative Commons licence CC BY-NC-SA 4.0.
v4.00, 2021. pdf file
v4.00, 2021 Source files