MATH W4041:
 Intro to Modern Algebra I
 Spring 2010

Week 7: March 2, 4     up     previous week     next week

Cayley Diagrams

Be comfortable drawing Cayley diagrams for all small groups. Here is my drawing for the quaternions (various students came up instead with 3D models). Glue opposite sides to form a torus, like playing "PacMan":

What makes this diagram tricky is that, unlike all others that we have seen, the faces don't tell the whole story. On a torus, there are closed loops that circle around the torus without bounding a face. (To be precise, there are two dimensions of such loops, and one computes this dimension in algebraic topology.) My diagram relies on these closed loops to identify vertices that would otherwise be distinct.

Burnside's Lemma

Wikipedia has a nice account (with proof) of Burnside's lemma. Be able to give this proof, and to apply Burnside's lemma to counting problems. (See last year's second midterm for some practice problems.)

Reading

Finish reading Chapter 5, with an emphasis on the later sections. Begin reading Chapter 6.

Exercises