Solutions to Quiz 2
1. Let R be a unital ring and : R S be a homomorphism such that (1) = 0. Show that (r) = 0 for all r R.
Let r R. Then
(r) = (1 r) = (1)(r) = 0(r) = 0.
2. Let R be a ring. Show that if x = x2 for all x R, then R has characteristic 2.
Le

Solutions for Assignment 2
#2 To show that SLn (R) is a subgroup, we need to prove that it is closed under matrix multiplication and contains inverses. If A, B SLn (R), then det(AB) = det(A) det(B) = 1,
so AB SLn (R). Likewise, if det(A) = 1, then det(A1

Assignment 5 Due Friday, October 10
Turn this in at the start of recitation on Friday, October 10.
1. Read Section 2.8 of Herstein.
2. Linear algebra and group theory
Vector spaces (with addition as their operation) and linear maps are good examples of gr

Assignment 4 Due Friday, October 3
Turn this in at the start of recitation on Friday, October 3.
1. Read Section 2.7 of Herstein. (Judson covers similar material in Chapter 11.)
2. Herstein, p. 53: 9
3. Let A1 be the group of ane functions on R. (In Assig

Solutions for Assignment 1
Part 1
2. (a) G = Z, a b = a b. Not a group. The operation isnt associative: (1 1) 1 =
0 1 = 1, but 1 (1 1) = 1 0 = 1.
(b) G = cfw_x Z | x > 0, with multiplication as operation. Not a group. 2 doesnt
have an inverse.
(c) Group.

Solutions for Assignment 3
p. 47, #5 Suppose that g G, so Hg is a right coset. Then
(Hg)1 = cfw_(hg)1 | h H
= cfw_g 1 h1 | h H g 1 H.
Furthermore, if g 1 h g 1 H, then (g 1 h)1 = h1 g Hg, so g 1 H (Hg)1 and
g 1 H = (Hg)1 .
Therefore, if X is a right coset

Solutions to assignment 4
p. 53: #9 If g G and H is a subgroup of G, we claim that gHg 1 is a subgroup of G. First, we show
closure: if ghg 1 , gh g 1 gHg 1 , then
(ghg 1 )(gh g 1 ) = ghg 1 gh g 1
= ghh g 1 gHg 1 .
Next, we show that it contains inverses:

Assignment 3 Due Friday, September 26
Turn this in at the start of recitation on Friday, September 26.
1. Read Section 2.52.6 of Herstein. (Judson covers similar material in Chapter 10.)
2. Herstein, p. 47, #5, 12 (consider the sets Hg1 Kg2 for g1 , g2 G)

Assignment 6 Due Friday, October 17
Turn this in at the start of recitation on Friday, October 17.
1. Read Section 2.9 of Herstein.
2. Suppose that H and N are normal subgroups of G, that H N = cfw_e, and HN = G.
(a) Show that for every g G, there are uni

Solutions to Assignment 5
#2
1. Addition in vector spaces is abelian, so every subgroup of Rn is normal. By the denition of vector
spaces, V is closed under addition and inverses, so it is a normal subgroup of Rn .
2. Elements of Rn /V are cosets of V of

Solutions to Assignment 10
1
Problem 1
Suppose that R is a commutative ring. Show that it satises a cancellation property
(i.e., that ab = ac implies b = c) if and only if it is an integral domain.
Solution. Suppose ab = ac for some a, b, c R and a = 0. W

a
F3
F2
b
c
F1
R0 = identity element (rotation by 0)
R1 = counterclockwise rotation by 2/3
R2 = counterclockwise rotation by 4/3
1
R0
R1
R2
F1
F2
F3
R0
R0
R1
R2
F1
F2
F3
R1
R1
R2
R0
F3
F1
F2
R2
R2
R0
R1
F2
F3
F1
F1
F1
F2
F3
R0
R1
R2
F2
F2
F3
F1
R2
R0
R

STUDY GUIDE
This is a guide to some of the skills, examples, and theorems that we've covered in class.
1.
Skills
Groups
Date
Dene group
Dene abelian
Decide whether a given set and operation form a group
Compute using group properties
Construct group

a
F3
F2
b
c
R0
R1
R2
F1
F2
F3
R0
R0
R1
R2
F1
F2
F3
R1
R1
R2
R0
F3
F1
F2
R2
R2
R0
R1
F2
F3
F1
F1
F1
F2
F3
R0
R1
R2
F1
The multiplication table of Sym()
F2
F2
F3
F1
R2
R0
R1
F3
F3
F1
F2
R1
R2
R0
a
F3
F2
b
c
R0
R1
R2
F1
F2
F3
R0
R0
R1
R2
F1
F2
F3
R1
R1
R2
R0

Review questions
December 10, 2014
1. Suppose that R is a Euclidean ring and that a, b R. We say that m R is a least common
multiple (lcm) of a and b if
a m and b m
If a m and b m , then m m
Let (a) = aR be the principal ideal generated by a.
(a) Show t

Assignment 7 Due Friday, October 31
Turn this in at the start of recitation on Friday, October 31.
Optional: Calculate the following products of permutations:
(1, 2, 3)(2, 4, 5) = (1, 4, 5, 2, 3)
(3, 4)(1, 2, 3, 4, 5, 6)(4, 3) = (1, 2, 4, 3, 5, 6)
(3, 7)

Solutions to Assignment 9
1. Which of the following sets are rings with respect to the usual operations of addition and multiplication?
If the set is a ring, is it also a eld?
(a) 4Z
4Z is an abelian group (it is a subgroup of Z). Furthermore, if 4a, 4b 4

Midterm review problems
These are a few problems to help you prepare for the midterm. You dont have to turn them in.
1. Suppose that H is a subgroup of G, that N is a normal subgroup of G, that H N = cfw_e, and HN = G.
(a) For all h H, let h : N N be the

Assignment 10 Due Friday, November 21
Turn this in at the start of recitation on Friday, November 21.
1. Suppose that R is a commutative ring. Show that it satises a cancellation property (i.e., that ab = ac
implies b = c) if and only if it is an integral

Assignment 11 Due Friday, December 5
Turn this in at the start of recitation on Friday, December 5.
1. Suppose that R and S are unital rings (not necessarily integral domains), that : R S is a
homomorphism, and that is onto. Prove that (1R ) = 1S .
2. Pro

Assignment 2 Due Friday, September 19
Turn this in at the start of recitation on Friday, September 19.
1. Read Sections 2.52.6 of Herstein. (Judson covers similar material in Chapters 3, 6, and 10.)
2. In recitation, you proved that the set of invertible

Assignment 8 Due Friday, November 7
Turn this in at the start of recitation on Friday, November 7.
1. Show that Z2 Z3 Z4 is not isomorphic to Z24 even though 2, 3, and 4 share no common factor.
2. List all of the nite abelian groups of order 36. Show that

Assignment 9 Due Friday, November 14
Turn this in at the start of recitation on Friday, November 14.
1. (Judson, p. 240, #1) Which of the following sets are rings with respect to the usual operations of
addition and multiplication? If the set is a ring, i

Assignment 1 Due Friday, September 12
Turn this in at the start of recitation on Friday, September 12.
Part 1
This part only uses what we covered in class on September 3. I recommend that you complete it before class
on September 8.
1. Read Chapter 1 and

Solutions to Assignment 6
#2
a) Let g G. Since HN = G, there is at least one h H and n N such that g = hn. We need to
show uniqueness. If hn = h n for h, h H and n, n N , then
hn = h n
1
(h )
hn = n
1
(h )
h = n n1 .
The left side of the equation is in H

Solutions to Assignment 11
1
Problem 1
Suppose that R and S are unital rings (not necessarily integral domains), then : R S is
a homomorphism, and that is onto. Prove that (1R ) = 1S .
Solution. Since is onto, for all s we can nd an element rs R so that (

Quiz 1 solutions
1. (a) What is a group?
A group is a set G with an operation that satises
Closure: a, b G a b G
Associativity: a, b, c G (a b) c = a (b c) G
Identity: there is an e G such that a e = e a = a for all a G
Inverses: for every a G, there is a

Midterm solutions
1. (a) What is a homomorphism?
A homomorphism between two groups G and H is a map f : G H such that f (g1 g2 ) =
f (g1 )f (g2 ) for every g1 , g2 G.
(b) Suppose that G is abelian and that f : G H is a surjective (onto) homomorphism.
Show

Final review solutions
1. Suppose that R is a Euclidean ring and that a, b R. We say that m R is a least common
multiple ( lcm) of a and b if
a m and b m
If a m and b m , then m m
Let (a) = aR be the principal ideal generated by a.
(a) Show that (a) (b)