# Math 817: Problem set 8

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, October 29, 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. Prove that {I, -I} is the unique normal subgroup of GL2(R) of order 2.
2. Let R denote the field of real numbers.
• (a) Prove that GL2(R) is not isomorphic to SL2(R) x (Rx). You may apply Problem 1, even if you do not turn that one in.
• (b) Prove that O2 is not isomorphic to SO2 x {I, -I}. [Hint: mimic your proof to part (a).]
3. Let A be an element of SO3 with angle a. Prove that cos(a) = (tr(A) - 1)/2.
4. Do #6.1, on p. 149 of Artin.
5. Do #6.5, on p. 149 of Artin. Assume the field C of complex numbers for part (b); for the other parts, assume the field R of real numbers.
6. A square matrix A is nilpotent if An = 0 for some positive integer n. If A is a nilpotent square matrix, prove that det(I + A) = 1.