Math 817: Problem set 11

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, November 19, 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. Each part of this problem might be useful for the next!
  2. Prove Proposition 1.12 on p. 199.
  3. Let G be the group of order preserving symmetries of the set X = {(a,0,1), (a,0,-1), (-a,1,0), (-a,-1,0)}, where a = 2-1/2. Note that X is the set of vertices of a regular tetrahedron T centered at the origin. Thus G is the tetrahedral subgroup of SO3. Determine the set S of poles of G (describe them explicitly, either in terms of T or by giving their coordinates), determine |S|, and explicitly determine the orbits of G acting on S (i.e., which poles are in which orbits). You do not need to justify your answers (except to yourself!).
  4. (Burnside's formula) Let G = {g1, ..., gr} be a group of order r acting on a finite set S. Let n be the number of orbits. For each g in G, let Sg = {s in S : gs = s}. Prove that n|G| = |Sg1| + ... + |Sgr|. [Hint: Look at C = {(g,s) : g is in G, s is in S, gs = s}.]
  5. (A generalization of Cayley's theorem) Let H be a subgroup of a group G of finite index [G:H] = n. Let G act on the left cosets G/H of H in G by left multiplication. Show there exists a homomorphism f: G -> Sn such that ker f is contained in H and every normal subgroup of G which is contained in H is contained in ker f (i.e., ker f is the largest normal subgroup of G contained in H).
  6. Let L = {me1 + ne2: m and n are integers}. Note that L is an additive subgroup of R2. Let v = (0.1, 0.01)t and let G = Sym(v + L) be the rigid motions of the plane which preserve the coset v + L.