Group Theory

This course is a proof-based introduction to group theory, one of the central organizing frameworks of modern mathematics. Students will study groups as abstract algebraic structures that encode symmetry, arithmetic, and transformation, and will learn how a small collection of axioms gives rise to rich and far-reaching theory.

We begin with the fundamentals: groups and finite groups, examples arising from number systems and geometry, subgroups, cyclic groups, and permutation groups. From there, students develop core structural tools including cosets, Lagrange’s Theorem, group homomorphisms and isomorphisms, and their consequences. Emphasis is placed on understanding why these results are true through careful proof, rather than on rote application.

As the course progresses, students explore deeper themes such as normal subgroups and factor groups, direct products, and selected applications to symmetry, counting, and geometry. Depending on student interest and readiness, topics may also include group actions, Burnside’s Lemma, symmetry groups, and selected results related to finite simple groups and the classification of small groups.

Throughout the course, students are expected to engage actively with proofs: constructing arguments, refining definitions, and learning how to communicate abstract ideas clearly and precisely. Problem sets emphasize reasoning, structure, and explanation over computation. Class meetings are discussion-based and collaborative, closely resembling a university-style seminar.

This course is ideal for students who are curious about abstraction, enjoy proof-based mathematics, and are interested in pursuing more advanced work in algebra, geometry, number theory, or mathematical research. It also serves as strong preparation for future research-oriented coursework and independent study within PiMath and beyond.