Need Help?

Need help finding the right class? Have a question about how classes work?

# Group Theory

Group theory is the study of symmetry. Objects in nature (physics, chemistry, music, etc.) as well as objects in mathematics itself have beautiful symmetries, and group theory is the algebraic language we use to unlock that beauty. Group theory is the gateway to abstract algebra and tells us (among many other things) that you can't trisect an angle with a straightedge and compass, that there are finitely many perfectly symmetric tiling patterns, and that there is no closed formula for solving a quintic polynomial. In this class we will get a glimpse of the mathematics underlying these famous questions. This course will focus concretely on building groups from other groups, exploring groups as symmetries of geometric objects, and using the tools of group theory to construct fields. The overarching goal of the course is to learn how modern mathematicians understand a topic as general and seemingly fuzzy as “symmetry”.

14 weeks

SYLLABUS
14 weeks

## Schedule

 Friday Jun 17 - Sep 16 7:30 - 9:30PM ET Jun 17 - Sep 167:30 - 9:30 PM Eastern6:30 - 8:30 PM Central5:30 - 7:30 PM Mountain4:30 - 6:30 PM PacificClick here to see more time zones Christian Hokaj \$520 (~\$38/week) \$520 (~\$38/week)

### AoPS Holidays

There are no classes May 28‐30, July 2‐4, September 3‐5, October 31, November 21‐27, and December 21‐January 3.

### Who Should Take?

This class is aimed primarily at students who have mastered the standard high school curriculum and do not have access to a strong post-secondary curriculum. We assume fluency with modular arithmetic, the complex numbers, and basic combinatorics, and also a good background in forming mathematical arguments and writing proofs. The class will be on the level of the most difficult Art of Problem Solving courses. We will not assume any calculus, but we will rely on precalculus, number theory, and counting extensively.

### Lessons

 Lesson 1 Symmetry Lesson 2 Examples of Groups Lesson 3 Subgroups Lesson 4 Abelian Groups Lesson 5 Group Actions Lesson 6 Orbits and Stabilizers Lesson 7 Burnside and Beyond Lesson 8 Quotients Lesson 9 Functions from Groups to Groups Lesson 10 Geometry and Group Theory Lesson 11 Fields Lesson 12 Field Extensions Lesson 13 Constructions and Automorphisms Lesson 14 Groups and Fields

I tried learning group theory on my own before taking this course and I had a pretty weak understanding. After taking this course, I have a clearer picture of group theory and how it works.