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Group TheoryGroup 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 |
| 14 weeks ARE YOU READY? SYLLABUS |
Schedule
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Join Mailing ListAoPS Holidays
There are no classes December 21 ‐ January 3, May 24 ‐ 26, July 4 ‐ 6, August 29 ‐ September 1, October 31, and November 24 ‐ November 30, 2025.
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
| 1 | Symmetry |
| 2 | Examples of Groups |
| 3 | Subgroups |
| 4 | Abelian Groups |
| 5 | Group Actions |
| 6 | Orbits and Stabilizers |
| 7 | Burnside and Beyond |
| 8 | Quotients |
| 9 | Functions from Groups to Groups |
| 10 | Geometry and Group Theory |
| 11 | Fields |
| 12 | Field Extensions |
| 13 | More Field Extensions and Geometric Constructions |
| 14 | Groups and Fields |
