Math 3124
Tuesday, November 1
Second Test. Answer All Problems.
Please Give Explanations For Your Answers.
1. In Z24 , prove that [4], [6] = [10] .
(10 points)
2. The following is part of the Cayley table for a group. Complete the table.
a
b
c
d
ab
b
cd
b
Math 3124
Thursday, October 20
October 20, Ungraded Homework
Problem 21.19 on page 109. True or false: if N G, then gng1 = n for all n N and
g G. Justify your answer.
False, e.g. (1 2 3) S3 , but (1 2)(1 2 3)(1 2)1 = (1 3 2).
Problem 21.26 on page 110. Pr
Math 3124
Tuesday, October 18
October 18, Ungraded Homework
Problem 19.3 on page 101 Determine whether the groups Z6 and S3 are isomorphic.
Z6 and S3 are not isomorphic because Z6 is abelian and S3 is nonabelian. Another reason is
that Z6 has an element o
Math 3124
Thursday, October 13
October 13, Ungraded Homework
1. Problem 21.4 on page 109. Dene a homomorphism r : M(2, Z) M(2, Z) by
10
r (x) = xr for each x M(2, Z), where r =
. Find ker r .
00
ker r = cfw_x M(2, Z) | xr = 0. Thus the matrix
ab
cd
ab
cd
Math 3124
Tuesday, October 11
October 11, Ungraded Homework
The starting position is
abcde
a
b
e
c
d
e
d
We are given bd = e and ec = d . Now which element is the
identity? The equality bd = e rules out b and d , and the equality ec = d rules out e and c.
Math 3124
Tuesday, October 4
October 4 Ungraded Homework
Problem 16.1 page 87
Determine the right cosets of [4] in Z8 .
[4] = cfw_[0], [4]. Now choose some element not in cfw_[0], [4], say [1]. Then [4] [1] =
cfw_[1], [5]. In the same fashion, we have [4]
Math 3124
Thursday, November 17
November 17, Ungraded Homework
1. Problem 39.1 on page 183. Dene : Z12 Z12 by ([a]12 ) = [a]4 for each [a]12
Z12 .
(a) Verify is well dened.
(b) Verify that is a homomorphism.
(c) Use the Fundamental Homomorphism Theorem f
Math 3124
Tuesday, November 15
November 15, Ungraded Homework
Problem 27.14 on page 135. What can be said about the characteristic of a ring R in which
x = x for each x R?
Since x = x for all x R, by adding x to both sides we see that 2x = 0 for all x R.
Math 3124
Thursday, November 10
November 10, Ungraded Homework
Problem 25.3 on page 127 Verify that ([2], [0]) is a zero divisor in Z3 Z3 .
Obviously ([2], [0]) = ([0], [0]). Also ([2], [0])([0], [2]) = ([0], [0]). Since ([0], [2]) = 0, this
shows that ([
Math 3124
Tuesday, November 8
November 8, Ungraded Homework
Problem 24.5 on page 124. Prove that Z[ 2] (that is cfw_a + b 2 | a, b Z) is a ring.
We have the two operations of addition and multiplication as usual, so associativity and
distributivity are OK
Math 3124
Thursday, November 3
November 3, Ungraded Homework
Let C = all cubes in (Z# , ) (that is cfw_x
211
by ( ) = 3 .
x
1. Prove that is a homomorphism and that C
x | x Z# ), and dene : Z# Z#
211
211
211
Z# .
211
2. Show that [14] ker .
3. Deduce that
Math 3124
Tuesday, October 25
October 25, Ungraded Homework
1. Problem 22.3 on page 114. Construct the table for Z12 / [4] .
Let us write k to indicate the coset to which k belongs. The order of [4] in Z12
is 3, so |Z12 / [4] | = 12/3 = 4. The elements of
Math 3124
Tuesday, August 23
Policy Sheet
Course
Prereq
Instructor
Book
Ofce
Telephone
E-mail
Room
Math 3124, Modern Algebra CRN (index number) 94246
Math 3034 (Introduction to Proofs)
Peter A. Linnell
Modern Algebra 6th edition, by John Durbin, ISBN 978-
Math 3124
Thursday, September 22
First Test Review
The test will cover as far as (and including) Section 14. Topics will include
1. Operations, associative and commutative law.
2. One-to-one and onto mappings. A mapping is invertible if and only if it is
Math 3124
Tuesday, September 27
First Test. Answer All Problems.
Please Give Explanations For Your Answers.
1. Which of the following formulae for x y denes an associative operation on Z# , the
4
set cfw_[1], [2], [3].
(a) x y = x
y
(b) x y = y
(10 points
Math 3124
Thursday, October 20
Sample Second Test. Answer All Problems.
Please Give Explanations For Your Answers.
1. Show that (1 2), (1 2 3 4) = (2 3 4), (1 3 4 2) .
(6 points)
2. The following is part of a Cayley table for a group. Complete the table.
Math 3124
Tuesday, September 13
Sample First Test. Answer All Problems.
Please Give Explanations For Your Answers.
1. Let S = cfw_1, 0, 1 and dene f : S S by f (x) = (x 1)x(x + 1) for x S, and
g : S S by g(x) = x3 .
(a) Determine which of f , g is onto.
(
Math 3124
Thursday, December 1
Sample Final Problems since Second Test
Important: the exam is comprehensive; the last third of the exam approximately will consist
of material covered since the second test. Here are some typical problems.
1. Let R be a rin
Math 3124
Thursday, September 29
September 29 Ungraded Homework
Problem 14.4 page 79 Determine the elements in each of the cyclic subgroups of Z6 .
Also give the order of each element of Z6 .
[0] [0]
[1] [0], [1], [2], [3], [4], [5]
[2] [0], [2], [4]
[3]
Math 3124
Tuesday, September 20
September 20 Ungraded Homework
Compute the order of
(1 2 3)(4 5 6 7), (1 2)(3 4 5 6 7 8)
in S8 S9 .
If a permutation is written as the product of disjoint cycles of lengths 1 , 2 , . . . , n , then the
permutation has order
Math 3124
Thursday, September 15
September 15 Ungraded Homework
Problem 14.6 page 79 Find the order of (1 2)(3 4 5) in S5 .
Since (1 2)(3 4 5) is already written as a product of disjoint cycles, this is [2,3] = 6 ([2,3]
means least the common multiple of
Math 3124
Tuesday, September 13
September 13 Ungraded Homework
Problem 11.14 page 65 Prove or disprove that Z# is a group with respect to .
4
# is not a group under . The reason is that closure for multiplication fails, so
Z4
is not an
operation. This is
Math 3124
Thursday, September 8
September 8 Ungraded Homework
Problem 8.5 on page 50 Determine the group of symmetries of a regular pentagon.
We will label the vertices of the regular pentagon 1, 2, 3, 4, 5 going clockwise. Denote by
A, B, C, D, E the lin
Math 3124
Tuesday, September 6
September 6 Ungraded Homework
8 10 7 9
6 12 5 11
. We rst move the space to the bottom right hand
The rst position is
4 14 3 13
2
1 15
corner; there will be several ways of doing this, and it doesnt matter which we choose. O
Math 3124
Thursday, October 27
Second Test Review
The test will cover sections 1522 excluding 20. RSA encryption will not be examined.
Topics will include
1. The subgroup generated by a subset.
2. Filling in Cayley tables.
3. Cosets.
4. Lagranges theorem.
Math 3124
Tuesday, November 15
Ninth Homework
Due 12:30 p.m., Thursday December 1
1. Let p be a prime and let R be an integral domain of characteristic p. Dene : R R
by (a) = a p . Prove that is a ring homomorphism. Show further that ker = 0.
(You may ass
Math 3124
Thursday, November 3
Eighth Homework
Due 12:30 p.m., Tuesday November 15
1. Let G = GL2 (Z5 ) and dene : G (Z# , ) by (A) = det(A).
5
(a) Prove that is a homomorphism of G onto Z# . You may assume the multiplicative
5
property for determinants o
Math 3124
Thursday, November 17
Eighth Homework Solutions
1. (a) (AB) = det(AB) = det(A)
det(B) = (A) (B), so is a homomorphism. It
x0
is onto because given x Z# , we have
= x, and the foregoing matrix is
5
01
in GL2 (Z5 ).
(b) A ker if and only if det(A
Math 3124
Thursday, October 13
Fifth Homework Solutions
1. Problem 14.33 on page 81. Prove or give a counterexample. If a group has a subgroup
of order n, then it has an element of order n.
S3 is a subgroup of S3 of order 6. But the elements of S3 have or