# Math 817: Problem set 1

Instructions: Write up any four of the following problems (but in the end you should be sure you know how to do all of them). Your write ups are due Friday, September 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.)
1. Translate the quote on p. 38. (A word for word translation is usually not a good translation.)
2. Let * be a composition law on a set S. Assume |S| = 2.
• (a) If S has an identity, e, under *, prove that * is both associative and commutative, but give an example to show that S still need not be a group.
• (b) If S has no identity element under * but * is commutative, must * be associative? Justify your answer (i.e., either give a proof that * is associative, or an example for which * is not associative).
3. If xyz = 1 in a group G, must yzx = 1? Justify your answer.
4. Note: it really would be enough just to do (b), as long as you show how (a) follows from your answer for (b).
• (a) If a and b are elements of a group G such that a3b = ba3, and a5 = 1, show that ab = ba.
• (b) State a more general fact that implies (a) as a special case. Prove your generalization.
• (a) Let H be a nonempty finite subset of a group G. Assume that ab is in H whenever a and b are in H. Show that H is a subgroup of G. [Hint: given x in H, consider the set S = {x, x2, x3, ... }.]
• (b) Is the hypothesis that H be finite necessary? Justify your answer.