2001- 02 Spring

MATH 466: Groups and Geometry

The following is what we have covered so far or what we will cover in the class.
• Week 1: (2 hours) Finite and infinite dihedral groups.
• Week 2: (2 hours) An orthogonal transformation of the Euclidean plane (R^2) is either a rotation at the origin or a reflection whose axis passes through the origin. An element of the Euclidean group (i.e. an isometry of the Euclidean plane) is a rotation or a reflection or a translation or a glide reflection.
• Week 3: (1 hour) The Euclidean group is generated by reflections. The combinations of various isometries were examined.
• Week 4: (3 hours) The Euclidean group is a semidirect product of the group of translations and the group of isometries that fix the origin. A finite subgroup of the Euclidean group is either cyclic or dihedral.
• Week 5: (3 hours + Exam 1) The symmetry groups of the Platonic solids were studied. A finite subgroup of SO3 is cyclic or dihedral or the symmetry group of one of the Platonic solids.
  The exam will cover the contents of the first 4 weeks only. Solving the questions of the first exercise sheet will be an excellent preparation for the exam.
• Week 6: (3 hours) Frieze groups were discussed and classified.
• Week 7: (2 hours) We proved that there are 5 types of lattices. A wallpaper group is defined to be a subgroup of the Euclidean group whose translation subgroup H is generated by two independent tranlsations and whose point group J is finite. It is possible to construct a lattice for a given wallpaper group by using the 'minimal generators' of H. Moreover J acts on this lattice. The order of a non-identity rotation in a wallpaper group is 2, 3, 4 or 6.
• Week 8: (3 hours) Some examples of wallpaper groups were studied. We started the classification of wallpaper groups.
• Week 9: (3 hours + Exam 2) The classification was completed.
• From Week 9 on, I follow the book by Humphreys. The sections that were covered from Chapter I are:
  1.1 - 1.9, 1.12 - 1.14 (inclusive).
  The proof of theorem 1.9 was postponed to the extra class.
• I shall cover 2.1 - 2.7 (and maybe 2.8 - 2.10 if time permits) from Chapter II.
• Exam 3. 16 May 2002, Thursday at 17:40 in M-05.
  Contents: 1.1 - 1.9, 1.12 - 1.14 (inclusive) and 2.1.