Title of the course: Dihedral groups
Instructor: Dr. Ulla Karhumäki
Institution: University of Helsinki
Dates: 22-28 July 2024
Prerequisites: First courses on algebraic structures (familiarity with groups, quotients, morphisms, etc.)
Level: Undergraduate (year 2-3)
Abstract: Let n ≥ 3. A dihedral group D_n is a group of symmetries of a regular n-gon, which includes rotations and reflections. Denote by r_n the element of D_n identified as a counterclockwise rotation by 2π/n radians and by s_n an element of D_n identified as a reflection across a line through a vertex. We first study the elementary aspects of dihedral groups D_n= <r_n, s_n>. Namely, we list their elements, relations between rotations and reflections, centers, and conjugacy classes, and show that the commutators in a dihedral group D_n form the subgroup <r^2>. After this, we give the abstract characterisation of D_n. Namely, we prove that for each group G =<x, y>, where x^n=1 for some n ≥ 3, y^2 = 1, and yxy =x^{-1}, there is a unique homomorphism from D_n to G sending r to x and s to y. Further, we describe the (normal) subgroups of D_n. We also introduce an infinite group that resembles the dihedral groups and has all of them as quotient groups. In the end of the course, we move on to study the symmetries of Platonic Solids.
Language: English