Math 116 First Midterm October 17, 2014

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1 Math 116 First Midterm October 17, 2014 Name: Instructor: Section: 1. Do not open this exam until you are told to do so. 2. This exam has 9 pages including this cover AND IS DOUBLE SIDED. There are 9 problems. Note that the problems are not of equal difficulty, so you may want to skip over and return to a problem on which you are stuck. 3. Do not separate the pages of this exam. If they do become separated, write your name on every page and point this out when you hand in the exam. 4. Please read the instructions for each individual problem carefully. One of the skills being tested on this exam is your ability to interpret mathematical questions. 5. Show an appropriate amount of work (including appropriate explanation). Include units in your answer where that is appropriate. Time is of course a consideration, but do not provide no work except when specified. 6. You may use no aids (e.g., calculators or notecards) on this exam. 7. If you use graphs or tables to find an answer, be sure to include an explanation and sketch of the graph that you use. 8. Turn off all cell phones and pagers, and remove all headphones and hats. 9. Remember that this is a chance to show what you ve learned, and that the questions are just prompts. Problem Points Score Total 110

2 Math 116 / Exam 1 (October 17, 2014) page 2 1. [10 points] Let G be a set, and let P = {p 1,...,p n } be a partition of that set (the p i are the parts of the partition). Let f : G G be any map, and define f(p i ) = {f(x) x p i } for any of the parts of P. Prove that f is a bijection if and only if f(p) = {f(p 1 ),...,f(p n )} is a partition of G.

3 Math 116 / Exam 1 (October 17, 2014) page 3 2. [8 points] on Z define by a b = a b. Does this make Z a monoid? Does it make a group? Why or why not? 3. [10 points] An element of a group S is idempotent if x x = x. Prove that a group has exactly one idempotent element.

4 Math 116 / Exam 1 (October 17, 2014) page 4 4. [12 points] Give the definition of a Monoid. Give an example of a Monoid that is not a Group. 5. [10 points] Given a group G and a subset H G. We know two different sets of conditions to test that H is a subgroup, state both of them.

5 Math 116 / Exam 1 (October 17, 2014) page 5 6. [8 points] Let f : Z Z/kZ be a map for some integer k. a. [4 points] Can f be injective? Why or why not? b. [4 points] Must f be surjective? Why or why not?

6 Math 116 / Exam 1 (October 17, 2014) page 6 7. [24 points] Let G = D 5 the group of symmetries of a pentagon. Let H = Z/10Z be the quotient group of Z by the normal subgroup 10Z. a. [6 points] Draw a pentagon with labeled corners, and give the set of elements in D 5. b. [6points] Draw thecayley graph of all subgroupsof G(hint: how bigcan those subgroups be?)

7 Math 116 / Exam 1 (October 17, 2014) page 7 c. [6 points] Is H a cyclic group? Why or why not? d. [6 points] Draw the Cayley graph of all subgroups of H.

8 Math 116 / Exam 1 (October 17, 2014) page 8 8. [23 points] Let G be a group of order 15. a. [5 points] Let g G. Can g have order 2? Why or why not? b. [6 points] Let g and g be two elements of G of order 5 with < g > < g > how big is < g > < g >?

9 Math 116 / Exam 1 (October 17, 2014) page 9 c. [6 points] Using the previous part, explain why G isn t entirely comprised of the identity and elements of order 5? d. [6 points] Assume you ve proved the previous section and know that G has an element whose order is neither 1 nor 5. Why can you conclude that G necessarily has an element of order 3? 9. [5 points] What is your favorite novel or biography?

Math 206 First Midterm February 1, 2012

Math 206 First Midterm February 1, 2012 Name: EXAM SOLUTIONS Instructor: Section: 1. Do not open this exam until you are told to do so. 2. This exam has 8 pages including this cover AND IS DOUBLE SIDED.