Group Theory - J.S. Milne   Top
Course Notes
Group Theory
Fields and Galois Theory
Algebraic Geometry
Algebraic Number Theory
Modular Functions and Modular Forms
Elliptic Curves -- see books.
Abelian Varieties
Lectures on Etale Cohomology
Class Field Theory
Complex Multiplication
Basic Theory of Affine Group Schemes
Lie Algebras, Algebraic Groups, and Lie Groups
Reductive Groups
Current version (3.14).
pdf file formatted for printing (11pt; a4paper; margins)
pdf file formatted for ereaders (9pt; 89mm x 120mm; 5mm margins)(3.11)

The first version of these notes was written for a first-year graduate algebra course. As in most such courses, the notes concentrated on abstract groups and, in particular, on finite groups. However, it is not as abstract groups that most mathematicians encounter groups, but rather as algebraic groups, topological groups, or Lie groups, and it is not just the groups themselves that are of interest, but also their linear representations. It is my intention (one day) to expand the notes to take account of this, and to produce a volume that, while still modest in size (c200 pages), will provide a more comprehensive introduction to group theory for beginning graduate students in mathematics, physics, and related fields.


  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


An undergraduate "abstract algebra" course.


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


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.