Math 817: Problem set 10

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 12, 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.)

Let n be an integer, at least 3.
- (a) Determine the center Z(D_n) of D_n. [Hint: Consider n even and n odd.]
- (b) Let G and H be groups. Prove that Z(GxH) = Z(G)xZ(H).
- (c) Prove that D_2n is isomorphic to D_nxZ(D_2n) if and only if n is odd.
Given a 2x2 integer matrix P, define a map T_P: ZxZ -> ZxZ by T_P(v) = Pv, where we regard elements of ZxZ as column vectors with integer entries.
- (a) Show that T_P is a homomorphism of groups.
- (b) Show that T_P is injective if and only if det(T_P) is not 0.
- (c) Show that T_P is surjective if and only if |det(T_P)| = 1.
- (d) Let h: ZxZ -> ZxZ be an isomorphism of groups. Show that h = T_P for some 2x2 integer matrix P of determinant 1 or -1.
Let G be a group and consider the set S = G. Define a map a: GxS -> S by a((g,s)) = gsg^-1, using the group multiplication and inversion.
- (a) Show that this defines an action of G on S = G (i.e., on itself).
- (b) Let h: G -> Perm(S) be the homomorphism associated to this action. Show that h(G) is a subgroup of Aut(G).
- (c) Show that ker h = Z(G).
Do Problem #10, on p. 190. For parts (a) and (b), it is enough to give just the answers; no justifications are needed. For part (a), your answer should be the complete list of all 2x2 matrices in the point group of G. For part (c), give a brief explanation of how you get your answer.
Do Problem #12 on p. 193.
Let G be a group acting on a set S. We say the action is trivial if gs = s for all s in S and all g in G. Now let H be a subgroup of G. Let H act on the left cosets of G/H by left multiplication; i.e., if h is in H and xH is in G/H, then h(xH) = (hx)H. Show that the action is trivial if and only if H is a normal subgroup of G.