Math 31  Homework 2 Solutions
1. [Saracino, Section 2, #1 (a), (b), (h), (i)] Which of the following are groups? Why? (That is,
either verify that the axioms hold, or explain why one of them fails.)
(a) R+ under addition. (Here R+ denotes the set of all
Challenge Problems!
Your nal bunch of challenge problems are due at the beginning of class on Monday, November 12.
1.) A fraction a/b Q is said to be reduced if gcd(a, b) = 1. Use the wellordering principle to
prove the following proposition: Every fract
Math 31 Homework 7
Note: This assignment is optional.
Note: Any problem labeled as show or prove should be written up as a formal proof, using
complete sentences to convey your ideas.
Basic Ring Theory
The problems on this list all involve basic denitions
Math 31  Homework 1
Due Friday, June 28
Easy
1.
Find gcd(a, b) and express gcd(a, b) as ma + nb for:
(a) (116, 84)
(b) (85, 65)
(c) (72, 26)
(d) (72, 25)
2.
Verify that the following elements of Zn , are invertible, and nd their multiplicative inverses.
Math 31  Homework 2
Due Wednesday, July 3
Easy
1. [Saracino, Section 2, #1 (a), (b), (h), (i)] Which of the following are groups? Why? (That is,
either verify that the axioms hold, or explain why one of them fails.)
(a) R+ under addition. (Here R+ denote
Math 31 Homework 6
Due Monday, August 12 (Changed from August 9)
Note: Any problem labeled as show or prove should be written up as a formal proof, using
complete sentences to convey your ideas.
Easier
1. We will see in class that the kernel of any homomo
Math 31 Homework 5
Due Friday, August 2
Note: Any problem labeled as show or prove should be written up as a formal proof, using
complete sentences to convey your ideas.
Easier
1. Determine if each mapping is a homomorphism. State why or why not. If it is
Math 31 Homework 4
Due Wednesday, July 17
Note: Any problem labeled as show or prove should be written up as a formal proof, using
complete sentences to convey your ideas.
Easier
1. Determine whether each of the following subsets is a subgroup of the give
Math 31  Homework 3
Due Wednesday, July 10
Note: Any problem labeled as show or prove should be written up as a formal proof, using
complete sentences to convey your ideas.
Easier
1. Let D4 be the 4th dihedral group, which consists of symmetries of the s
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Math 31 Homework 6 Solutions
1. We will see in class that the kernel of any homomorphism is a normal subgroup. Conversely,
you will show that any normal subgroup is the kernel of some homomorphism. That is, let G be a
group with N a normal subgroup of G,
Math 31 Homework 5 Solutions
Easier
1. Determine if each mapping is a homomorphism. State why or why not. If it is a homomorphism,
nd its kernel, and determine whether it is onetoone and onto.
(a) Dene : Z R by (n) = n. (Both are groups under addition h
Math 31 Problem Set # 6 Solutions
3. If R is a nite integral domain, show that R is a eld.
To show that R is a eld, we must show that every nonzero element of R is a unit. Let r R
be nonzero. Consider the innite sequence of ring elements r1 , r2 , r3 , .
Math 31 Problem Set # 6
Due Wednesday, October 31 (Halloween)
This is your rst problem set after your very dicult midterm. As usual, turn in two disjoint solution
sets. The rst should contain your writeups for problems 1.) through 5.), and the second sho
Math 31 Problem Set # 5 Solutions
1.) A simple group is one whose only normal subgroups are the trivial group and the
whole group. Let G be a simple group and suppose : G H is a group homomorphism. Prove that is either the trivial homomorphism or injectiv
Math 31 Problem Set # 5
Due Wednesday, October 17
This is your last problem set before your very dicult midterm. As usual, turn in two disjoint solution
sets. The rst should contain your writeups for problems 1.) through 5.), and the second should contai
Math 31 Problem Set # 4 Solutions
1.) Write the following permutations in cycle notation.
(a) =
123456
413526
(b) =
= (1452)(3)(6)
123456
312564
= (132)(456)
(c) =
123456
654321
= (16)(25)(34)
2.) Let , , and be as in the rst problem.
(a) Compute , , ,
Math 31 Problem Set # 4
Due Wednesday, October 10
This problems set is longer than your previous ones, but it also easier. Please turn in two disjoint solution
sets. The rst should contain your solutions for problems 1.) through 6.), and the second should
Math 31 Problem Set # 3
Due Wednesday, October 3
Please turn in two disjoint solution sets. The rst should contain your solutions for problems 1.) through
5.), and the second should contain your solutions for 6.) through 10.). Be sure each is stapled and
Math 31 Problem Set # 3 Solutions
1.) Check whether the relation is an equivalence relation on the given set S :
(a) S = R and x y if x y is rational.
This can be seen as a special case of the following theorem from class: Suppose H G. Then
the relation x
Math 31 Problem Set # 2
Due Wednesday, September 26
For grading purposes, please turn in two disjoint solution sets. The rst should contain your solutions
for problems 1.) through 5.), and the second should contain your solutions for 6.) through 10.). Be
Math 31 Problem Set # 1
Due Wednesday, September 19
For grading purposes, please turn in two disjoint solution sets. The rst should contain solutions
to problems 1.) through 4.), and the second should contain solutions to 5.) through 8.). Be sure
each is