Math 817: Problem set 2

Instructions: Write up any four of the following problems (but you should try all of them, and in the end you should be sure you know how to do all of them). Your write ups are due Friday, September 10, 2004. Each problem is worth 10 points, 9 points for correctness and 1 point for communication. (Your goal is not only to give correct answers but to communicate your ideas well. Make sure you use good English, so proofread your solutions. Once you finish a solution, you should restructure awkward sentences, and strike out anything that is not needed in your approach to the problem.)
  1. Let a = 5723 and let b = 5959.
  2. For this problem, we start with some terminology. Let G be a group. Let g be an element of g; then we have a mapping fg: G -> G called conjugation by g, defined by fg(x) = gxg-1, for all x in G. An element of the form gxg-1 is called the conjugate of x by g. The center of G, denoted Z(G), is the subset {x in G : xy = yx for all y in G} of G.
  3. Do problem 7 on p. 71: in the notation on p. 11, let A be the 2x2 matrix I2 + e1,2 and let B = I2 + e2,1. Show that A is conjugate to B (or, equivalently, vice versa) in GL2(R), but they are not conjugate in SL2(R).
  4. Let G = (g) be a finite cyclic group of order n.
  5. Let f: X -> Y be a mapping of sets. Let A be a subset of X and let B be a subset of Y.