Math 55A

Math 55A is an intensive honors algebra course for first-year Harvard College students, providing a rigorous introduction to abstract algebra, including group theory and linear algebra. This course covers the equivalent of Math 25a and Math 122, and prepares students for Math 123 and other advanced courses in number theory and algebra.

Course Information

Course Name: Studies in Algebra and Group Theory

Course Number: Math 55A

Instructor: Denis Auroux

Course Description:

This course provides a rigorous introduction to abstract algebra, including group theory and linear algebra. The formal prerequisites for Math 55 are minimal, but this class does require a commitment to a demanding course, strong interest in mathematics, and some familiarity with proofs and abstract reasoning.

Course Content:

This course covers the following main topics:

• Group theory: groups, subgroups, homomorphisms, quotient groups, symmetric groups
• Linear algebra: vector spaces, linear maps, matrices, eigenvalues and eigenvectors
• Standard forms of linear operators: Jordan normal form, characteristic and minimal polynomials
• Bilinear forms: inner products, orthogonality, spectral theorem
• Tensor products: tensor algebra, symmetric and exterior algebras
• Module theory: modules over rings, classification of finitely generated abelian groups
• Group actions: orbits, stabilizers, Burnside’s lemma, conjugacy classes
• Sylow theorems and applications
• Group representation theory: representations of finite groups, irreducible representations, character theory

Textbooks:

Required Texts:

• Linear Algebra Done Right, Sheldon Axler
• Algebra, Michael Artin

Other Recommended References:

• Abstract Algebra, David S. Dummit, Richard M. Foote
• Naive Set Theory, Paul Halmos
• Finite-Dimensional Vector Spaces, Paul Halmos
• Representation Theory: A First Course, William Fulton, Joe Harris
• Linear Representations of Finite Groups, Jean-Pierre Serre

Course Outline:

• Weeks 1-5: Fundamentals of group theory (groups, subgroups, homomorphisms, quotient groups, symmetric groups)
• Weeks 6-12: Linear algebra (vector spaces, linear maps, eigenvalues, Jordan normal form)
• Week 13: Midterm 1
• Weeks 14-20: Bilinear forms, inner products, spectral theorem, tensor products
• Weeks 21-26: Module theory, group actions, Sylow theorems
• Week 27: Midterm 2
• Weeks 28-37: Group representation theory (representations, irreducible representations, character theory)

Prerequisites: The formal prerequisites for Math 55 are minimal, but this class does require a commitment to a demanding course, strong interest in mathematics, and some familiarity with proofs and abstract reasoning. Students should be familiar with general set-theoretic notation and language, as well as basic proof techniques.

Note: This course information is compiled based on Harvard University course content.