Math 417 Homework 6

Math 417 Homework 6: Due Friday March 7

Instructions: You can discuss these problems with others, but write up your solutions on your own (i.e., don't just copy someone else's solutions, else the feedback I give you won't help you much). Please be neat and write in full sentences.
Do four of the eight problems, the four you didn't do on the exam.

[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 : Subsets(B) -> Subsets(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\ge 2 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 > 0.
[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 be a fixed positive integer. Let the subset N of G be the set of all elements g in G such that gⁿ = e. Prove that N is a subgroup of G.
[6]
- (a) Write (1 2 3 4 5 6)³ as a product of disjoint cycles.
- (b) Write (1 2 3)(1 2)(3 4) as a product of disjoint cycles.
- (c) Find |(1 2 3)(1 2)| and show how you find your answer.
[7] Consider the group Z₈₉₉ of integers modulo 899. Note that 899 = 29(31). 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 we have ab in H, but nonetheless H is not a subgroup of G.