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ABSTRACT ALGEBRA
V Semester
CORE COURSE
For
B. Sc. MATHEMATICS
(CUCBCSS -2014 admn.onwards)
UNIVERSITY OF CALICUT
SCHOOL OF DISTANCE EDUCATION
Calicut University P.O. Malappuram, Kerala, India 673 635
354-A
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UNIVERSITY OF CALICUT
SCHOOL OF DISTANCE EDUCATION
STUDY MATERIAL
For
B. Sc. MATHEMATICS
V Semester
CORE COURSE
ABSTRACT ALGEBRA
Prepared by :Sri. Aboobacker PAssistant Professor
Department of MathematicsW M O College, Muttil
Scrutinised by: Dr.D.Jayaprasad,Principal,Sreekrishna College Guruvayur,Chairman,Board of Studies in Mathematics(UG)
Layout:
Computer Section, SDE
©Reserved
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CONTENTS PAGES
CHAPTER - 1 05-26
CHAPTER - II 27-42
CHAPTER - III 43-50
CHAPTER - IV 51-58
CHAPTER - V 59-69
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CHAPTER 1INTRODUCTION TO GROUPS
The axioms for a group are short and natural. . . .Yet somehow hidden behind these axioms is themonster simple group, a huge and extraordinary
mathematical object, which appears to rely on numerousbizarre coincidences to exist. The axioms
for groups give no obvious hint that anything likethis exists.
Richard Borcherds, in Mathematicians 2009
IntroductionMathematics is often referred to as a science — sometimes as thescience of patterns.You will enjoy your studies of mathematics more if youlook for patterns in all the ideas you explore. One such idea is that of groups.Group theory is applied to many areas of science such as genetics,quantum theory, molecular orbits, crystallography and the theory of relativity.In mathematics, group theory is applied to many mathematical modelsinvolving algebra, number theory and geometry.In chapter 1 you dealt with different sets of numbers within the RealNumberSystem. Throughout your student life you have used the operations ofaddition, multiplication,subtraction and division, finding a square root,reciprocals, and so on. These are examples of operations performed onnumbers that are part of a certain set. Operations (such as addition) thatinvolve 2 input values, for example 2+3, are called binary operations . Thosethat involve only one input value, such as finding the square root of a number(for example ) are calledunaryoperations. Others that involve 3 input values are called ternary ; forexample, the principal, interest and term of a loan are the 3 input valuesinvolved in calculating the amount of interest due on a loan. (Strictly speakingthe multiplication involved is still carried out on pairs of values)
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of positive integers is not closed under subtraction ,because 2,3 is in Z+ ,but 2-3=-1 is not in Z+.
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Problem
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Problem
Solution
Problem
An operation * is defined as a*b =ab+a+1. Is * commutative? Is *associative?
Solution
a*b= ab+a+1 and b*a=ba+b+1 ,ie a*b≠ b*a.
Therefore * is not commutative.
( a*b)*c = (ab+a+1)*c =(ab+a+1)c + ab+a+1+c = abc+ab+ac+a+2c+1
a*(b*c)=a*(bc+b+1) =a(bc+b+1)+a+1 = abc+ab+2a+1
ie ( a*b)*c ≠ a*(b*c)
Therefore * is not associative.
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Structural properties
a)The operation is commutative
b) Cardinality of the sets
c)The number of solution of x*x =c
Example
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Example
Example
Remark
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Example
Problem
Solution
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Problem
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Exercise
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1.2 Groups: Definition and Elementaryproperties
Examples
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Reversal Law
Problem
Theorem1.2.3 Cancellation Laws
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Proof
If possible x1 and x 2 are the solution of a*x =b.
Then a* x1 =b and a* x2 =b
Hence a* x1 = a* x2 implies x1 = x2 (By Left cancellation law)
Try the other yourself !
Problem
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Problem
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Problem
Problem
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Examples
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(In the problem 4 the operation is +4, the addition modulo 4. a+4 b= r, the reminderobtained when a+b is divided by 4 )
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Problem
Problem
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Examples
Problem
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Problem
Problem
Problem
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Chapter 2GROUPS OF PERMUTATIONS
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Example
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Example
Problem
Problem
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Problem
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Example
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Problem
Problem
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Example
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Problem
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Example
Problem
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Chapter 3COSETS AND THE THEOREM OF LAGRANGE
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Problem
Problem
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Example
Problem
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Problem
Problem
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Chapter 4HOMOMORPHISM
Example
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Example
Problem
Problem
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3.
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Problem
Problem
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Problem
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Chapter 5
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Example
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Example
Example
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Example
Example
Example
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Problem
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Example
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Example
Problem
Problem
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Problem