# 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!
• (a) Let Z be the center of a group G. If G/Z is cyclic, show that G is abelian (and hence G=Z).
• (b) If G is a nonabelian group of order 10, show that the class equation for G must be 10 = 1 + 2 + 2 + 5.
• (c) If G is a nonabelian group of order 10, show that G is isomorphic to D5. [Hint: conclude from (b) that G has an element x of order 5 and an element y of order 2. Show that yxy = x4, and show that this uniquely determines the multiplication table of G. Hence, up to isomorphism, there is a unique nonabelian group of order 10.]
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.
• Determine the point group of G (actually list the matrices explicitly; no proof is needed).
• Determine the translation subgroup LG of G (no proof is needed, but be clear as to which vectors are in LG).
• Artin shows that the point group acts on LG (see p. 169), but goes on to say that ``it is important to note that ... G need not operate on LG.'' Verify this by exhibiting an element g of G and a vector w in LG, such that gw is not in LG.