Math 817: Problem set 4

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 24, 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 and N be subgroups of a group G, with N normal in G. Let I be the intersection of H and N. You may assume without proof the fact that I is normal in H and N is normal in HN (but of course you should be able to prove this fact if you need to!). The Second Isomorphism Theorem states that: H/I is isomorphic to HN/N. Prove the Second Isomorphism Theorem. [Hint: Define an appropriate homomorphism F: H -> HN/N and apply the First Isomorphism Theorem.]
Let K and N be normal subgroups of a group G, with N contained in K. Use the FFF to show that K/N is a normal subgroup of G/N, and prove the Third Isomorphism Theorem (AKA the Freshman's Dream), that (G/N)/(K/N) is isomorphic to G/K. [Hint: Apply the FFF and the First Isomorphism Theorem.]
Let g: Z -> G and h: Z -> H be surjective homomorphisms of groups, where Z denotes the integers. Let |G| = m and |H| = n. Let F : Z -> GxH denote the homomorphism whose components are g and h.
- (a) Find ker F and |Im(F)|. Justify your answers.
- (b) If gcd(m, n) = 1, show that Z/mnZ is isomorphic to GxH. [Hint: apply (a), using the First isomorphism Theorem.]
- (c) Given positive integers m and n such that gcd(m, n) = 1 and arbitrary integers a and b, prove the Chinese Remainder Theorem (CRT), that there is an integer c which is a simultaneous solution x = c to x = a mod m and x = b mod n (I'm using "=" since the usual triple equals is not really available in html), and that the set of all solutions is precisely the set of integers d such that mn | c - d. [Hint: Restate the CRT in terms of F.]
Let m and n be relatively prime positive integers. Given any integer i, let [i]_m denote the coset i+mZ in Z (thus [i]_m is just the congruence class of i modulo m).
- (a) You may assume that the map g: Z/mnZ -> Z/mZ x Z/nZ defined by g([i]_mn) = ([i]_m, [i]_n) is an isomorphism. What is the inverse? (I.e., give a formula for where ([i]_m, [j]_n) must map to, given arbitrary integers i and j. You do not need to give proof, but be sure your answer really is the inverse.)
- (b) You may assume that the map h: Z/mZ x Z/nZ -> Z/mnZ defined by h(([i]_m, [j]_n)) = [ni+mj]_mn is an isomorphism. What is the inverse? (I.e., give a formula for where [i]_mn must map to, given an arbitrary integer i. You do not need to give proof, but be sure your answer really is the inverse.)
This is #8, p. 75: if G is a group with normal subgroups of orders 3 and 5, prove that G has an element of order 15.
(This is essentially #10.1, p. 76.) Let G be the group of invertible real upper triangular 2x2 matrices. For each of the following conditions, determine if it defines a normal subgroup H of G (it is enough to just write "Yes" or "No" without justification), and if so, determine G/H (it is enough without justification to define an explicit surjective homomorphism G -> K to some group K such that the kernel is H; by the First Isomorphism Theorem, this would show that G/H is isomorphic to K, just be sure that your map really is an isomorphism with the required kernel!).

(a) a₁₁ = 1 (b) a₁₂ = 0 (c) a₁₁ = a₂₂ (d) a₁₁ = a₂₂ = 1