MAT 534 Spring 2014

The course is over! Thanks to everyone for a great semester, and enjoy your summer.

Basic Info


Prof. Graham Leuschke
Time & Place
MWF 9:30–10:25 in Carn 115
Joseph A Gallian, Contemporary Abstract Algebra, 8th ed.
textbook website
Office Hours
MW 11-12, TR 2-3; by appointment; and any time my door is open (317G Carnegie)
Important documents


  • The third midterm exam will be in class on Friday, April 25. It will cover Chapters 10 through 14 (up through ideals and factor rings).
  • The second midterm exam was in class on Friday, March 28. It covered up through Chapter 9 (normal subgroups and factor groups).
  • The first midterm exam was Friday, February 21, as in the syllabus. The exam covered up through Chapter 6 (Isomorphisms).


  • Why all rings should have a 1, by Bjorn Poonen
  • Solutions to some of the homework: hw03, hw04, hw05
  • Keeler’s theorem and products of distinct transpositions: An episode of Futurama features a two-body mind-switching machine which will not work more than once on the same pair of bodies. After the Futurama community engages in a mind-switching spree, the question is asked, “Can the switching be undone so as to restore all minds to their original bodies?” Ken Keeler found an algorithm that undoes any mind-scrambling permutation with the aid of two “outsiders.” We refine Keeler’s result by providing a more efficient algorithm that uses the smallest possible number of switches.
  • Permutations and the 15-puzzle from class on 3 Feb. Skip to Section 5 for the good stuff, then come back to Section 4 for the details.
  • Group theory and Rubik’s Cube from class on 3 Feb. Skip to Section 3. There are about 519 quintillion conceivable configurations for the Rubik’s Cube, but only 1/12 of them are “valid”.

Problem Sets