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.
 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.
 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.
 Let {f_{1}, f_{2}, ... }
be a sequence of integers such that
f_{1} > 1 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 >= 1.
 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}.)
 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
g^{n} = e.
Prove that N is a subgroup of G.

 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.
 Consider the group Z_{899}
of integers modulo 899. Note that
899 = 29x31. For each positive integer n, determine the number of
elements of Z_{899} of order n.
Explain how you obtain your answer.
 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.