MATH 466 (Spring 2006)
2005 - 06 Spring
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.