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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






Summer 2023This course will be offered in Summer 2023. Click here to join our mailing list to be notified when the course schedule is available.

AoPS Holidays

There are no classes May 27‐29, July 4, September 2‐4, October 31, November 20‐27, and December 21‐January 3, 2024.

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.


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

This is the first time I've gotten into higher math, and I think this class is the best possible way to do it. It's been my favorite AoPS class so far.

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