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errata 2005/8/23 15:21 page 1 #1
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Errata
Goodman, Algebra: Abstract and Concrete, 2nd ed.
Page 6: Comments on the paragraph following gure 1.2.5: The
centroid of the square is the center of mass; it is the intersection of
the two diagonals.
Consid
Chapter 1
Group Fundamentals
1.1
1.1.1
Groups and Subgroups
Denition
A group is a nonempty set G on which there is dened a binary operation (a, b) ab
satisfying the following properties.
Closure : If a and b belong to G, then ab is also in G;
Associativit
Abstract Algebra done Concretely
Donu Arapura
February 19, 2004
Introduction: I wrote these notes for my math 453 class since I couldnt
nd a book that covered basic abstract algebra with the level and emphasis that
I wanted. Rather than spending a lot of
Notes on Abstract Algebra
John Perry
Current address: University of Southern Mississippi
E-mail address: john.perry@usm.edu
URL: http:/www.math.usm.edu/perry/
Copyright 2009 John Perry
www.math.usm.edu/perry/
Creative Commons Attribution-Noncommercial-Sha
METU, Fall 2010, Math 111, Section 1.
Homework 1
1. Prove the distributive law
P (Q R) (P Q) (P R)
by using the full truth tables of both statements.
2. Prove the new variable law
P (P Q) (P Q)
using other laws of logic. Justify each step.
3. What is the
Math367 Abstract Algebra Fall 2014
Quiz 5
Name:
Surname:
Student No:
Question 1.
a) Is Z3 Z9 isomorphic to Z27 ? Why?
Solution: Z3 Z9 is not isomorphic to Z27 . Since Z27 is a cyclic group of order
27, it has an element of order 27.
On the other hand, for
Math367 Abstract Algebra Fall 2014
Quiz 6
Name:
Surname:
Student No:
Question 1. Let R be a ring and a, b R. Show that for any positive integer n,
(na)b = n(ab) = a(nb).
Solution: Let n be a positive integer.
Then (na)b = (a + + a)b = ab + + ab = n(ab).
a
Math 236
Fall 2006
Dr. Seelinger
Solution for 3.1
Problem 5: Consider the following sets and determine whether each set is a subring of M2 (R).
If a set is a subring of M2 (R), determine whether it has an identity.
0 r
(a) Let S be the set of all matrices
Arkansas Tech University
MATH 4033: Elementary Modern Algebra
Dr. Marcel B. Finan
25
Integral Domains. Subrings
In Section 24 we defined the terms unitary rings and commutative rings.
These terms together with the concept of zero divisors discussed below
Ring Theory Problem Set 1 Solutions
Problems 1a,b,c,e from the text.
SOLUTION: All of these examples are rings as explained in class. For example, 7Z is an
additive subgroup of Z which is clearly closed under multiplication and hence 7Z is a subring
of Z.
Michael Barr Charles Wells
Toposes, Triples and Theories
Version 1.1 10 September 2000
Copyright 2000 by Michael Barr and Charles Frederick Wells. This version may be downloaded and printed in unmodified form for private use only. It is available at http:
page 1 of Frontmatter
Abstract Algebra: The Basic Graduate Year Robert B. Ash PREFACE This is a text for the basic graduate sequence in abstract algebra, offered by most universities. We study fundamental algebraic structures, namely groups, rings, fields
TeAM YYePG
Digitally signed by TeAM YYePG DN: cn=TeAM YYePG, c=US, o=TeAM YYePG, ou=TeAM YYePG, email=yyepg@msn.com Reason: I attest to the accuracy and integrity of this document Date: 2005.01.23 16:28:19 +08'00'
ABEL'S THEOREM IN PROBLEMS AND SOLUTIONS
Jonathan Bergkno
Herstein Solutions
Chapters 1 and 2
Throughout, G is a group and p is a prime integer unless otherwise stated. A B denotes that A is a
subgroup of B while A
B denotes that A is a normal subgroup of B .
H 1.3.14* (Fermats Little Theorem) P
Math 367 Abstract Algebra Fall 2014
Quiz 9
Name:
Surname:
Question 1. Show that h6, 10, 15i = Z.
Solution:
Let I = h6, 10, 15i.
As I is an ideal of Z, I Z. Need to show Z I.
Since I is an ideal, 6 + 10 + 15(1) = 1 I.
Let z Z. Then z = z.1 I as 1 I .
Hence
Math367 Abstract Algebra Fall 2014
Quiz 3
Name:
Surname:
Id:
Question 1. If a group G has a unique element x of order 2, then show that
x Z(G). (Hint: Consider the order of g 1 xg for any g G.)
Solution:
Let g be an arbitrary element of G. Then
(g 1 xg)2
Math 367 Abstract Algebra Fall 2014
Quiz 8
Name:
Surname:
Student No:
Question 1. Prove that Z[x] is not a principal ideal domain.
(You can consider the ideal hx, 2i)
Solution:
Consider the ideal I = hx, 2i). We will show that I is not principal.
Assume t
Oct 27, 2010
METU, Fall 2010, Math 111, Section 1.
Quiz 1
1. Negate and simplify
x P (x) y Q(y).
The final expression should involve only the symbols , and . Justify each step.
2. Are these statements true or false? Explain your answers briefly. The unive
METU
Department of Mathematics
Abstract Algebra
1
Code
Acad. Year
Semester
Instructor
: Math 367
: 2015
: Fall
: K
ucu
ksakall
Date
Time
Duration
: Nov 9, 2015
: 17:40
: 100 minutes
2
3
4
5
Midterm 1
Last Name :
:
Name
Student No. :
:
Signature
6 QUESTION
2215 Solutions Problem Sheet 11
Ring, Subrings, Ideals, Intregal Domains, Ring Homomorphisms, First
Isomorphism Theorem.
1. Let R, S be rings. Verify that the direct product, that is,
R S = cfw_(a, b) : a R, b S
with operations
(a, b) + (c, d) = (a + c, b
Math367 Abstract Algebra Fall 2014
Quiz 2
Question 1. Let f : G1 G2 be a homomorphism and let N E G2 . Show that
f 1 (N ) is a normal subgroup of G1 .
Solution: Firstly, we need to show that f 1 (N ) is a subgroup of G1 .
Since f is a homomorphism we hav
METU
Department of Mathematics
Abstract Algebra
1
Code
Acad. Year
Semester
Instructor
: Math 367
: 2015
: Fall
: K
ucu
ksakall
Date
Time
Duration
: Dec 14, 2015
: 17:40
: 120 minutes
2
3
4
5
Midterm 2
Last Name :
:
Name
Student No. :
:
Signature
6 QUESTIO
Math367 Abstract Algebra Fall 2014
Quiz 1
Question 1. Let G be a group and let H = cfw_g G | gx = xg x G. Show that H
is a subgroup of G.
Solution: Since 1x = x1 x G, 1 G so H 6= . Let g, h H. Then gx = xg
and hx = xh x G. As hx = xh we have xh1 = h1 x x
METU
Department of Mathematics
Abstract Algebra
1
Code
Acad. Year
Semester
Instructor
: Math 367
: 2015
: Fall
: K
ucu
ksakall
Date
Time
Duration
: Jan 19, 2016
: 13:30
: 135 minutes
2
3
4
5
Final Exam
Last Name :
:
Name
Student No. :
:
Signature
8 QUESTI
Math 367 Abstract Algebra Fall 2014
Quiz 7
Name:
Surname:
Student No:
Question 1. Prove that every ring homomorphism from Zn to itself has the
form (x) = ax where a2 = a.
Solution: Let be a ring homomorphism from Zn to itself. Let (1) = a
Then a = (1) = (
METU
Department of Mathematics
Abstract Algebra
1
Code
Acad. Year
Semester
Instructor
: Math 367
: 2015
: Fall
: K
ucu
ksakall
Date
Time
Duration
: Nov 9, 2015
: 17:40
: 100 minutes
2
3
4
5
Midterm 1
Last Name :
:
Name
Student No. :
:
Signature
6 QUESTION
CENKER OZAN AKMAN 1913011
METU nformatic nstitute
Introduction to
Information
Technologies and Its
Application
Alan Turing
Table of contents
1
Alan Turing
1.1 Early Life and Education.1
1.2
Scientific Career.2
1.2.1
1.2.2
1.2.3
1.3
Best Known Papers.2
Cra
Nov 24, 2010
METU, Fall 2010, Math 111, Section 1.
Quiz 2
1. If A = cfw_1, 2, 3, then give an example of a relation R from A to A with 4 elements.
Find R1 .
Find R R1 .
2. Define F = cfw_(x, y) R R : |x| = |y|.
x R(y R(xF y). True or false? Explain.
I
Math367 Abstract Algebra Fall 2014
Quiz 4
Name:
Surname:
Student No:
Question 1. Prove that G is abelian if and only if the commutator group of G
is trivial.
Solution:
Assume that G is abelian. Take an arbitrary generator x1 y 1 xy of [G, G], where
x, y G
Solutions to Assignment 3
Question 1. [Exercises 3.1, # 2]
Let R = cfw_0, e, b, c with addition and multiplication defined by the following tables. Assume associativity
and distributivity and show that R is a ring with identity. Is R commutative? Is R a f