## 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 {f
_{1}, f_{2}, ...} be a sequence of integers such that
f_{1}\ge 2 and for i > 0 such that f_{i+1} >= 2f_{i} - 1.
Prove that f_{n} >= (2^{n}+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} = gh^{i}g^{-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 g
^{n} = 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
**Z**_{899} of integers modulo 899. Note that
899 = 29(31). For each positive integer n, determine the number of
elements of **Z**_{899} 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.