MATH 466 (Spring 2006)

# MATH 466: Groups and Geometry

The picture of the last lecture (12 May 2006) with Coxeter graphs and baklavas!

Objects in METU with large finite symmetry groups

Announcements:
• Final exam: 26 May 2006, Friday, 13.30, M-102.
• VERY IMPORTANT: Students changed the grading system, see below.
• IMPORTANT: Change in schedule: We are meeting on Tuesdays at 12.40-13.30 in M-106, instead of Wednesdays, starting from 21 Februray on.
• If you are a student taking MATH 466, then you can subscribe the e-mail group.
• Visiting the webpage of Math 466 from four years ago may give you an idea about the course.

### Lectures:

Note the change: Tuesday 12.40-13.30 (M-106), Friday 13.40-15.30 (M-105)

### Books:

For the contents of the first 3 weeks, I shall give you handouts. For the next 3 weeks, I shall loosely follow "Groups and Symmetry" by M. A. Armstrong. Then we will follow "Reflection Groups and Coxeter Groups" by James E. Humphreys.
You can find a list of related books here.

### Exams:

Midterm: 11 May 2006, Thursday, 17.40. Contents: From frieze groups to section 2.1 of Humphreys.
Final: 26 May 2006, Friday, 13.30, M-102. Contents: 1.1-1.9 and 2.1-2.7 from Humphreys.

### Grading:

Homework - 20 points
Midterm - 30 points
Final - 50 points
Those who do not attend at least 30% of the lectures and do get a failing grade in the exams will receive NA as their final letter grades.

### In the class:

Week 1: The set of symmetries of a subset of Rn forms a group under composition. Various properties of the symmetry groups of regular n-gons (aka finite dihedral groups) were investigated. In particular, we proved that they are isomorphic to the semi-direct product of a finite cyclic group with Z2.

Week 2: Sym(Z) (the inifinite dihedral group) and Sym(R) were studied. Also we proved that there are exactly 5 Platonic solids and examined them.

Week 3: The symmetry groups of Platonic solids were discussed. In particular, we proved that the symmetry groups of the terahedron, octahedron and the icosahedron are isomorphic to S4, S4 x Z2, and A5 x Z2, respectively. Also orthogonal linear transfromations were discussed.

Week 4: We showed that SO2 (and SO3) consists of rotations of the plane (and the space resp.) at the center and O2\SO2 consists of reflections whose axes pass through the origin. We classified finite subgroups of O2. We proved some basic theorems about group actions.

Week 5: With the help of the theorems on group actions, we classified the finite subgroups of O3.

Week 6: We showed that Isom(R2) has 4 types of elements; rotations, translations, reflections and glide reflections. We studied the isometry group of Rn. In particular, we showed that it is isomorphic to the semi-direct product of Rn and On. We examined the compositions of different types of isometries in Isom(R2) and showed that reflections generate Isom(R2).

Week 7: We will study the frieze groups and more.

This line last updated on 23 May 2006.