Intro to Group Theory

Introduction to Group Theory: As devices for measuring symmetry, groups occupy a central role in several areas of mathematics. This course begins with the definition of a group and finishes with the Sylow Theorems for finite groups and the Fundamental Theorem of Finite Abelian Groups. Emphasis is placed on teaching through examples.

Lectures
Prerequisites

To do the problem sets: some proofs, but heavy on example-oriented problems. For those looking for a more functional and less theoretical understanding of finite groups, the basics of integers and primes should be enough.

To follow the lectures: Basic knowledge of proofs, set theory, and the integers. The focus will be on finite groups, with the occasional infinite example. Most linear algebra exercises are labeled advanced; we use mostly 2x2 matrices, with occasional 3x3 cases.

Base list: Sets, bijections/one-one correspondences, equivalence relations, and equivalence classes. Factorization of integers into primes. A video playlist for the prerequisites can be found here.

Syllabus

Week 1. Definition of Group

Week 2. Subgroups

Week 3. Cosets and Lagrange’s Theorem

Week 5. Special Subgroup Constructions

Week 6. Homomorphisms and Isomorphisms

Week 7. Non-Isomorphic Groups/ Automorphisms

Week 8. Fermat's Little Theorem

Week 9. Groups of Order 8/ Semidirect Products

Week 10. Group Actions/ Cayley's Theorem

Week 11. Symmetric Groups/ Conjugacy and The Class Equation

Week 12. Cauchy's Theorem/ Sylow Theory

Week 13. Internal Products/ The Fundamental Theorem of Finite Abelian Groups

Week 14. Composition and Classification (Last Class)

Office Hours: I'll be running the course out of the subreddit GroupTheory2012. Questions and comments can be posted there.

Videos: The complete video list is available in the Abstract Algebra section, GT series at mathdoctorbob.org.

Book: I'm using Herstein's Topics in Algebra for an outline. He doesn't do group actions, and it is not a great book for beginners. I'm a big fan of Schaum's Outlines for beginners. Dixon's Problems in Group Theory is also affordable with many solved problems.