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 R^{n} 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 Z_{2}.
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 S_{4},
S_{4} x Z_{2}, and A_{5} x Z_{2},
respectively. Also orthogonal linear transfromations were discussed.
Week 4: We showed that SO_{2} (and SO_{3}) consists
of rotations of the plane (and the space resp.) at the center and
O_{2}\SO_{2} consists of reflections whose axes pass
through the origin. We classified finite subgroups of O_{2}. 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 O_{3}.
Week 6: We showed that Isom(R^{2}) has 4 types of
elements; rotations, translations, reflections and glide reflections. We
studied the isometry group of R^{n}. In particular, we showed
that it is isomorphic to the semi-direct product of R^{n} and
O_{n}. We examined the compositions of different types of
isometries in Isom(R^{2}) and showed that reflections generate
Isom(R^{2}).
Week 7: We will study the frieze groups and more.
This line last updated on 23 May 2006.