The study of the symmetries of an object leads naturally to the definition of an abstract group. The course alternates between studying the symmetries of specific mathematical objects and building the necessary vocabulary and theory of abstract groups. The course begins with the study of rigid motions of the Euclidean plane. We characterise the isometries (distance preserving maps) of the plane and introduce the notion of groups in this context. We then proceed to study important examples and properties of groups. The course alternates between the abstract theory of groups and the study of symmetry in the context of Euclidean geometry.
We study the structure theory of groups via the algebraic notions of order of an element, group homomorphisms, subgroups, cosets of subgroups, Lagrange’s theorem, normal subgroups, quotient groups and isomorphism theorems. The course also introduces the notion of group acting on a set, and the Orbit-Stabiliser Theorem and some of its applications. Alongside this theory, the course explores the symmetry groups of subsets of the plane (including Frieze patterns, and wallpaper patterns, rigid motions of ℝ3, the orthogonal group and its subgroups, and the symmetry groups of platonic solids.