Groups and Symmetry



Basic Information
Instructor Prof. Chris  Leininger
411 Mathematics
Phone:  854-2431
clein@math.columbia.edu
Office hours
Mon. 10:30am--12:00noon & Thur. 9:00am--10:30am and by appointment (all held in Math 411 unless otherwise stated).
TA's
Sonja  Mapes -- Tue. 10am--1pm in Milbank help room
Cristina Caputo-- Tue. 3pm--4pm in Milbank help room
Class times Mon. & Wed.: 9:10am -- 10:25am in Math 207
Text Groups and symmetry; A Guide to Discovering Mathematics, David W. Farmer;
Mathematics of the Rubik's cube, W. D. Joyner (download here; see unpublished (without figures but smaller file) or here with figures (larger file)).
Other stuff
Rubik's cube (available online or e.g. at Toy's 'R Us).
Topics The topics I have in mind to cover are: groups, subgroups, generators, Lagrange's Theorem, Euler's Theorem and cryptography, normal subgroups, homomorphisms, isomorphisms, Isomorphism Theorems, products, Cayley's Theorem, Permutation groups, group actions, semi-direct products, puzzles (esp. 15-puzzle and Rubik's cube), SL_{2}(Z), Braid groups. This list is merely a guide. I don't know if we'll get to even half the things listed here or if we'll go beyond them. Also, we will not necessarily discuss these topics in this order.
Midterm 1 October 6
Midterm 2 November 17
Final exam T.B.A.
Exam details
There will be no make up exams.
The exams will test you on your understanding of the concepts covered to that point in class.  In particular, you will be required to come up with proofs of facts not yet discussed, but which follow from those that have been.
There will be no practice exams.  You may consider the homework as practice for the exams.
In an attempt to remove the possibility of cheating, there will be up to 8 different versions of each exam.
Grading
Each exam will count as 1/3 of your grade.
homwork details
The logistics of homework in a class of 150 students is a little tricky. On the other hand, working on and solving problems is a crucial part of doing mathematics. Furthermore, discussing mathematics with your peers is also of great importance (and very often overlooked by students). The plan for homework is as follows: The class should divide up into teams of 4--6 students each and arrange at least one 1-hour block of time per week to discuss the homework. Likely, the best days to do this are Thursday--Sunday, as homework will be assigned Monday and/or Wednesday. You should first try to figure out the problems on your own. Then, come to your team meetings with your solutions as well as any questions you have about the homework. Don't be discouraged; these meetings can be difficult, even if (or especially if) your teammates are your friends. Each team will turn in one, two, or three problems every monday. These will be written neatly by one of the team members (take turns: each member should write up the homework at least twice throughout the semester). These will be looked over by myself or one of the T.A.'s and returned by Wednesday's class. You will not receive a grade for your homework -- these are for your benefit -- but I'll keep track of who's turning reasonable work in.
The homework problems will each be assigned a point value; 1, 2, or 3. The teams should decide on the problems to turn in, and submit up to 3 points worth of problems. E.g. you can turn in three 1-point problems or one 1-point problem and a 2-point problem, etc. The points are a gauge of how long it will likely take me to look at your solution.
Homework update! none.
Useful Links Columbia math home page,
Mathematics of the Rubik's cube (1 source),
Mathematics of the Rubik's cube (2nd source), more later...
Supplemetary notes
"Final" set of revised notes (.ps format) (.pdf format)
Generators for layer subgroups
of the Rubiks cube group(.ps)
  (.pdf)
homework-
Assignment 1a (.ps) (.pdf) Solutions, 1a (.ps) (.pdf)
Assgn. 1 b (.ps) (.pdf) Solutions, 1b (.ps) (.pdf)
Assgn. 2 (.ps) (.pdf) (see 5th notes for definition of order)Solutions, 2 (.ps)(.pdf)
Assgn. 3 (.ps) (.pdf) Solutions, 3 (.ps) (.pdf)
Assgn. 4 (.ps)  (.pdf)   some solutions to Assgn. 4 & 5 (.ps)
Assgn. 5 (.ps)  (.pdf)                                           (.pdf)
Assgn. 6 (.ps)  (.pdf)   solutions to Assgn. 6 (.ps)   (.pdf)
Assgn. 7 (.ps)  (.pdf)
Assgn. 8 (.ps)  (.pdf)   solutions to Assgn. 8 (.ps)   (.pdf)
Assgn. 9 (.ps)  (.pdf)   solutions to Assgn. 9 (.ps)   (.pdf)
Assgn. 10 (.ps)  (.pdf)  solutions to Assgn. 10 (.ps)  (.pdf)