# Math 817: Problem set 2

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 10, 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. Let a = 5723 and let b = 5959.
• (a) Find a generator for the subgroup aZ + bZ of Z.
• (b) What is the least positive integer c such that there is a solution (x, y) in integers for the equation ax + by = c? Justify your answer, and find a solution (x, y) for that c.
2. For this problem, we start with some terminology. Let G be a group. Let g be an element of g; then we have a mapping fg: G -> G called conjugation by g, defined by fg(x) = gxg-1, for all x in G. An element of the form gxg-1 is called the conjugate of x by g. The center of G, denoted Z(G), is the subset {x in G : xy = yx for all y in G} of G.
• (a) Let Aut(G) denote the set of all automorphisms of G. Show that Aut(G) is a subgroup of the group Perms(G) of all permutations of G (hence Aut(G) is a group).
• (b) The book shows that fg is an automorphism. Show that a: G -> Aut(G) defined by a(g) = fg is a homomorphism with kernel Z(G) (hence Z(G) is a normal subgroup).
• (c) If H is a subgroup of G, then fg(H) is traditionally denoted gHg-1, called the conjugate of H by g. Since gHg-1 is the image of H under the homomorphism fg, we know that gHg-1 is a subgroup of G. Show that H is normal if and only if gHg-1 = H for all g in G.
3. Do problem 7 on p. 71: in the notation on p. 11, let A be the 2x2 matrix I2 + e1,2 and let B = I2 + e2,1. Show that A is conjugate to B (or, equivalently, vice versa) in GL2(R), but they are not conjugate in SL2(R).
4. Let G = (g) be a finite cyclic group of order n.
• (a) Let i be a positive integer. Show that (gi) = (gd), where d = gcd(i, n), and that |gi| = n/d.
• (b) Show that there is a one to one correspondence between positive integer divisors of n and subgroups of G. [Hint: Justify the fact that every subgroup of G is cyclic and apply (a).]
• (c) Show that a finite group H is cyclic if and only if there is a unique cyclic subgroup of H of order j for each divisor j > 0 of |H|.
5. Let f: X -> Y be a mapping of sets. Let A be a subset of X and let B be a subset of Y.
• Show that f(f-1(B)) is a subset of B, and that f(f-1(B)) = B if f is surjective.
• Formulate and prove an analogue of (a) involving injectivity.