M417 Practice Exam 1 Spring 2004

Note: This is the Exam 1 I gave when I taught this class last year. We had gotten a little further; our Exam 1 next Monday won't cover anything beyond Chapter 4 (so you don't have to worry about a problem like # 6 below). Like last year's exam, this year's will have a mix of problems, some right from the homework, some similar to homework problems or things done in class, and some new ones, and you will get a choice. I will post answers later this week.

Instructions: Do any four of the eight problems. Don't forget to put your name on your answer sheets.

1. Let h: A -> B be a function from a set A to a set B. If h: A -> B is surjective, show that h^-1 : 2^B -> 2^A is injective.
2. Let G be a group which has exactly three different subgroups, including a proper subgroup H of order 7. Show that G is cyclic, and determine |G|.
3. Let {f₁, f₂, ... } be a sequence of integers such that f₁ > 1 and for i > 0 such that f_i+1 >= 2f_i - 1. Prove that f_n >= (2ⁿ + 2)/2 is true for all integers n >= 1.
4. Let g and h be elements of a group G. Note that G need not be finite. Prove that |ghg^-1| = |h|. (Hint: show that (ghg^-1)ⁱ = ghⁱg^-1.)
5. Let G be an abelian group. Let n > 0 be an integer. Let N be the subset of G of all elements g of G such that gⁿ = e. Prove that N is a subgroup of G.
6. • Write (1 2 3 4 5 6)³ as a product of disjoint cycles.
   • Write (1 2 3)(1 2)(3 4) as a product of disjoint cycles.
   • Find |(1 2 3)(1 2)|; show how you find your answer.
7. Consider the group Z₈₉₉ of integers modulo 899. Note that 899 = 29x31. For each positive integer n, determine the number of elements of Z₈₉₉ of order n. Explain how you obtain your answer.
8. Give an example of a group G and a nonempty subset H of G such that whenever a is in H and b is in H, then ab is in H, but nonetheless H is not a subgroup of G.