Math 3140 Fall 2012
Exam #1
Work alone. No materials except pen (or pencil) and paper allowed.
Write your solutions on a separate paper. Justify your answers. Giving
incorrect or irrelevant justication will be penalized.
Problem 1. Show that the function
Math 3140 Fall 2012
Assignment #1
Due Weds., Sep. 5.
Exercise 1. (a) List all of the symmetries of a square, allowing all transformations made up of rotations, reections, and translations. This group is
called D4 .
Solution. Label the vertices 1, 2, 3, an
Math 3140 Fall 2012
Exam #3
Due Wednesday, December 19. Work alone. You may consult any
textual references you want (including textbooks and the internet),
but you may not discuss the exam with anyone (neither in person, nor
over the phone or internet) ex
Math 3140 Fall 2012
Exam #2
Work alone. No materials except pen (or pencil) and paper allowed.
Write your solutions on a separate paper. Justify your answers. Giving
incorrect or irrelevant justication will be penalized.
Problem 1. Suppose that A, B , and
Math 3140 Fall 2012
Exam #1
Work alone. No materials except pen (or pencil) and paper allowed.
Write your solutions on a separate paper. Justify your answers. Giving
incorrect or irrelevant justication will be penalized.
Problem 1. Show that the function
Math 3140 Fall 2012
Exam #2
Work alone. No materials except pen (or pencil) and paper allowed.
Write your solutions on a separate paper. Justify your answers. Giving
incorrect or irrelevant justication will be penalized.
Problem 1. Suppose that A, B , and
Math 3140 Fall 2012
Assignment #2
Due Fri., Sept. 14.
sources.
Remember to cite your
Exercise 2. [Fra, 4, #9]. Let U be the set of complex numbers of absolute
value 1.
(a) Show that U is a group under multiplication of complex numbers.
Solution. First we
Math 3140 Fall 2012
Assignment #3
Due Fri., Sept. 21. Remember to cite your
sources, including the people you talk to.
My solutions will repeatedly use the following proposition from class:
Proposition 1. Let G be a group and H G a subset. If H is non-emp
Math 3140 Fall 2012
Exam #1 Rejected problems
Exercise 1. Verify that if : A B and : B C are homomorphisms of groups then : A C is also a
homomorphism.
Exercise 2. Prove that an abelian group G and a non-abelian group H cannot be isomorphic to each other.
Math 3140 Fall 2012
Assignment #7
Due Mon., Oct. 29.
sources.
Remember to cite your
Exercise 1. List the cosets of each of the following subgroups:
(a) 8Z Z
Solution.
8Z, 1 + 8Z, 2 + 8Z, 3 + 8Z, 4 + 8Z, 5 + 8Z, 6 + 8Z, 7 + 8Z
(b) 9Z/36Z Z/36Z
Solution.
9Z
Math 3140 Fall 2012
Assignment #4
Due Fri., Sept. 28. Remember to cite your
sources, including the people you talk to.
In this problem set, we use the notation gcd cfw_a, b for the greatest common
divisor of a and b.
My solutions will use the following pr