Introduction to Group Theory

ABOUT THE COURSE: Group theory studies symmetries. This course, designed for undergraduate students, gives a gentle introduction to the highlights of elementary group theory. The course begins with group definitions and basic properties and discusses various essential examples of groups from Geometry and Number theory. We move on to developing basic notions like homomorphisms and quotient groups. Finally, we study group actions and see various applications of group actions, including Sylow’s theorems.PREREQUISITES: Basic Linear Algebra

Week 1: Groups: definitions and basic propertiesWeek 2:Examples from Geometry I: Group of isometries of a plane and their finite reflection subgroupsWeek 3:Examples from Geometry II: symmetries of a cube, tetrahedron and regular n-gonWeek 4:Examples from Number theory: the additive/multiplicative group of integers modulo nWeek 5:Cyclic groups: various characterizationsWeek 6:Abelian groups examples, Symmetric and alternating groupsWeek 7:Group homomorphisms, Normal subgroupsWeek 8:Quotient groups and isomorphism theoremsWeek 9:Group actions: definition and various examplesWeek 10:Application of group action: Cayley’s theorem, Lagrange’s theorem, Cauchy's theoremWeek 11:Conjugacy action, the class equationWeek 12:Sylow’s theory and its application