Math 817: Problem set 12

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, December 3, 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 H be a subgroup of prime index p in a finite group G. Use the generalized Cayley's theorem to prove that H is normal, if p is the smallest prime dividing the order of G.
Let G be the automorphism group of D₅.
- (a) Determine |G|.
- (b) Show that G is not abelian.
- (c) Find an element f of G of order 4.
- (d) In the terminology of Problem 2 from Homework 2, show that f² = a(g) for some g in D₅.
- (a) Find a representative of each conjugacy class of A₅, and determine the order of the class. (No justification is needed.)
- (b) Determine the orders of possible normal subgroups of A₅.
- (c) Use your answer to (b) to determine whether A₅ is a simple group.
Do Problem 8 on p. 196.
Let G be a finite group whose order is divisible by a prime p. The point of this problem is to prove Cauchy's theorem, that G must have an element of order p. So let the cyclic group C_p = {1, c, c², ..., c^p-1} act on the direct product GxGx...xG = G^p (of p copies of G) by cyclic permutations; i.e., if (g₁, ..., g_p) is an element of G^p, then, for example, c(g₁, g₂, ..., g_p) = (g_p, g₁, ..., g_p-1). Let U = {(g₁, g₂, ..., g_p) in G : g₁g₂...g_p = 1_G} - {(1_G, ..., 1_G)}; i.e., U is the subset of G^p of all sequences of p elements of G (other than the constant sequence whose terms are all the identity) such that the product of the terms is the identity.
- (a) Show that U is a union of orbits under the action of C_p on G^p, hence that C_p acts on U.
- (b) Show that p doesn't divide |U|.
- (c) Use the Fixed Point Theorem (p. 199) to show that G has an element of order p. [Hint: Show that an element (g₁, g₂, ..., g_p) of U is fixed under the action of C_p if and only if all of the entries g_i are equal.]