Group Theory Online Math Course
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Group Theory

Group theory is the study of symmetry. Objects in nature (math, physics, chemistry, etc.) have beautiful symmetries and group theory is the algebraic language we use to unlock that beauty. Group theory is the gateway to abstract algebra which is what tells us (among many other things) that you can't trisect an angle, that there are finitely many regular polyhedra, and that there is no closed form for solving a quintic. In this class we will get a glimpse of the mathematics underlying these famous questions. This course will focus specifically on building groups from other groups, exploring groups as symmetries of other objects, and using the tools of group theory to construct fields.

18 weeks


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AoPS Holidays

There are no classes November 25–December 1, December 21–January 3, May 23–25, July 3–5, and September 5–7.

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 Cyclic Groups
Lesson 4 Abelian Groups
Lesson 5 Group Actions I
Lesson 6 Group Actions II
Lesson 7 Quotients
Lesson 8 Functions from Groups to Groups
Lesson 9 Group Presentations
Lesson 10 Symmetric and Alternating Groups
Lesson 11 Group Classification and Sylow's Theorems
Lesson 12 Examples of Fields
Lesson 13 Building Finite Fields
Lesson 14 Polynomials
Lesson 15 Vector Spaces and Dimension
Lesson 16 Number Fields and Constructions
Lesson 17 Galois Groups of Finite Fields
Lesson 18 Automorphisms of Number Fields and Solving Quintics

Group theory was definitely a wildly new and unexplored branch of mathematics to me, but thanks to AoPS's problem-solving approach to the subject, I gained a very firm foundation in this daunting subject. I appreciated the thoughtful clarity in the teaching in the class, and the knowledgeable, thorough instructor and teaching assistant really made this a great course. Thank you very much for this unique tour through advanced mathematics!

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