Math 481, Abstract Algebra I, Winter 2012
Homework Solutions
1. Characterize those integers n such that any Abelian group of order n belongs to one of exactly four
isomorphism classes.
Solution. The integers n which have this property are those of the for
1. In how many ways can 5 boys and 3 girls be seated in a round table so that no girls
are together?
Solution:
There are 5 boys. So, there are (51)! ways to arrange them in a circular table.
Now, we have to place 3 girls in 5 gaps so that no girls are tog
Dihedral Group
The dihedral group of order , denoted by , consists of the six symmetries of an equilateral
triangle. The elements that comprise the group are three rotations: , , and
counterclockwise about the center of , , and , respectively; and three r
MECHANICAL ENGINEERING DEPTT.,
DELHI TECHNOLOGICAL UNIVERSITY, BAWANA ROAD, DELHI-42.
NOTICE
CHOICES FOR SPECIALISATION OF AREA IN MECHANICAL ENGINEERING:
All the students of VIIth semester, B.E.(Mechanical Engg) are required to submit their
options for s
Solutions of Practice Test 3 MA407H
1. Let R =
ab
cd
| a, b, c, d Z
a0
0a
and let S =
|aZ .
(a) (10 points) Prove or disprove that S is a subring of R.
(b) (10 points) Prove or disprove that S is an ideal of R.
Solution: (a) Let
a0
0a
a0
0a
a
0
and
a
0
0
MAS 4301
Selected Exercises from Chapter 14
31) Let R be the ring of continuous functions on R, and let A = cfw_f R : f (0) = 0. Prove
that A is a maximal ideal.
Proof: First, we know that A is an ideal, because if f and g are in A, then (f g )(0) = f (0)
Math 482, Abstract Algebra 2, Spring 2004
Problem Set 2, solutions
13.20 Find all units, zero-divisors, idempotents, and nilpotent elements in Z3 Z6 .
An element (a, b) is a unit if there is (c, d) such that (a, b)(c, d) = (ab, cd) = (1, 1). This can
occu
Math 481, Abstract Algebra, Winter 2004
Problem Set 8, due Tuesday, March 9
10.12 Suppose that k is a divisor of n. Prove that Zn / k Zk .
=
Let : Zn / k Zk be the map given by (x + n ) + k + n
this map is an isomorphism.
x + k . We will show that
First,
College of the Holy Cross, Fall 2009 Math 351, Practice Final: Hints Prof. Jones
1. Suppose a group contains elements a and b such that |a| = 4, |b| = 2 and a3 b = ba. Find |ab|. Remember that the order of an element x G is the smallest positive integer n
0701/160.302
ALB
Internal Only
CP1
MASSEY UNIVERSITY
ALBANY CAMPUS
EXAMINATION FOR
160.302 ALGEBRA
Semester One 2007
Time Allowed: THREE (3) hours
Answer ALL (16) questions
CP1
Non-programmable calculators only are permitted.
THIS IS A CLOSED BOOK EXAMINA
CBSE TEST PAPER-03
CLASS - IX Mathematics (Polynomials)
1.
2.
3.
4.
The value of K for which x 1 is a factor of the polynomial 4x3 + 3x2 4x + K is
(a) 0
(b) 3
(c) 3
(d) 1
The factors 12x2 x 6
[1]
[1]
(a) (3x 2) (4x + 3)
(b) (12x + 1) (x 6)
(c) (12x 1) (x
CBSE TEST PAPER-02
CLASS - IX Mathematics (Polynomials)
1.
2.
3.
4.
Which of the following expression is a monomial
(a) 3 + x
(b) 4x3
(c) x6 + 2x2 + 2
(d) None of these
A linear polynomial
[1]
[1]
(a) May have one zero
(b) has one and only one zero
(c) Ma
Life Sciences
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SUNIL GUPTA*
The author reflects on the research process, the review process, and the past and
future research related to his 1988 article, which won the 1993 O'Dell award.
Reflections on "Impact of Sales Promotions on
When, What, and How Much to Buy'
The
PART I : MENTAL ABILITY TEST
Time : 90 Minutes
s.c
o
Questions: 1 to 100
m
MODEL TEST PAPER 1 (MAT & SAT)
Directions: (Q. 1 to 7)
In each of the following questions, a number series is given which is in accordance with
tic
some certain rule. What is the c
Chapter - 2
(Polynomials)
Key Concepts
Constants : A symbol having a fixed numerical value is called a constant.
Example : 7, 3, -2, 3/7, etc. are all constants.
Variables : A symbol which may be assigned different numerical values is known as
variable.
E
CBSE TEST PAPER-05
CLASS - IX Mathematics (Polynomials)
1.
2.
If x+y+x = 0, then x3 + y 3 + z 3 is
(a) xyz
(b) 2xyz
4.
(c) 3xyz
(d) 0
The value of (x-a)3 + (x-b)3 + (x-c)3 3 (x-a) (x-b) (x-c) when a + b + c= 3x, is
(a) 3
3.
[1]
(b) 2
(c) 1
(d) 0
Factors o
CBSE TEST PAPER-04
CLASS - IX Mathematics (Polynomials)
1.
2.
3.
4.
The value of 1023 is
(a) 1061208
(c) 1820058
[1]
(b) 1001208
(d) none of these
(a-b)3 + (b-c)3 + (c-a)3 is equal to
(a) 3abc
(b) 3(a-b) (b-c) (c-a)
(c) 3a3b3bc3
(d) [a-(b+c)]3
The zeroes
CBSE TEST PAPER-01
CLASS - IX Mathematics (Polynomials)
1.
Which of the following expression is a polynomials
(a) x3 1
(b)
x +2
1
x2
(d)
t + 5t 1
(c) x 2
2.
3.
4.
A polynomial of degree 3 in x has at most
(a) 5 terms
(b) 3 terms
(c) 4 terms
(d) 1 term
Th
Homework #1 Solutions
Chapter 1
Practice Problems 1. With pictures and words, describe each symmetry in D3 (the set of symmetries of an equilateral triangle). There are six symmetries: Three rotations of 0 , 120 , and 240 , and ips over three dierent axes
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Section 11 Direct Products
and Finitely Generated Abelian Groups
One purpose of this section is to show a way to use known groups as
building blocks to form more groups.
Definition:
The Cartesian product of sets S1, S2, ,Sn is the set of all ordered ntupl
Math 103A Homework Solutions
HW7
Jacek Nowacki
March 12, 2007
Chapter 10
Problem 2. Prove that the mapping : R R , dened by (x) = |x|,
is a homomorphism with Ker() = cfw_1, 1.
Proof. The above map is dened by the absolute value. In order to prove
that it
MATH 103A HW 9 SOLUTIONS (NONAUTHORITATIVE)
Chapter 6 Problems
Problem 6: We wish to nd Aut(Z). So suppose is an automorphism of Z. Then is determined
by (1). To see this, suppose n Z, then since is a homomorphism, (n) = (n 1) = n(1).
Now note that since
IMMERSE 2010
Algebra Course
Problem Set 5 Solutions
1. Let G be a group and Z (G) be the center of G. Prove that if G/Z (G) is cyclic, then G is abelian.
Proof (by Matthew, Lilith, Derrek, Linda). Since G/Z (G) is cyclic, then lets pick some generator of
Math 481, Abstract Algebra, Winter 2004
Problem Set 7, due Friday, February 20
1. Prove that Ut (st) U (s).
=
Let be the map from Ut (st) to U (s) given by x mod st x mod t. We must show that this
map is well dened, a homomorphism, injective, and surjecti
Math 481, Abstract Algebra, Winter 2004
Problem Set 6, due Friday, February 13
6.31 Show that the mapping a log10 a is an isomorphism from R+ under multiplication to R
under addition.
Lets call the map in question . Now is obviously well dened (there is n