ABSTRACT ALGEBRA

ABSTRACT ALGEBRA

V Semester

CORE COURSE

B. Sc. MATHEMATICS

(CUCBCSS -2014 admn.onwards)

UNIVERSITY OF CALICUT

SCHOOL OF DISTANCE EDUCATION

Calicut University P.O. Malappuram, Kerala, India 673 635

School of Distance Education

Abstract Algebra Page 2

UNIVERSITY OF CALICUT

SCHOOL OF DISTANCE EDUCATION

STUDY MATERIAL

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

CONTENTS PAGES

CHAPTER - 1 05-26

CHAPTER - II 27-42

CHAPTER - III 43-50

CHAPTER - IV 51-58

CHAPTER - V 59-69

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)

of positive integers is not closed under subtraction ,because 2,3 is in Z+ ,but 2-3=-1 is not in Z+.

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.

Structural properties

a)The operation is commutative

b) Cardinality of the sets

c)The number of solution of x*x =c

Example

Remark

Example

Problem

Solution

Problem

Exercise

1.2 Groups: Definition and Elementaryproperties

Examples

Reversal Law

Problem

Theorem1.2.3 Cancellation Laws

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

Examples

(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 )

Problem

Examples

Problem

Chapter 2GROUPS OF PERMUTATIONS

Example

Problem

Example

Problem

Example

Problem

Example

Problem

Chapter 3COSETS AND THE THEOREM OF LAGRANGE

Problem

Example

Problem

Chapter 4HOMOMORPHISM

Example

Problem

Chapter 5

Example

Problem

Example

Problem

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