MTH-376

Algebra

Lecture 1

Instructor: Dr. Muhammad Fazeel Anwar

Assistant Professor

Department of Mathematics

COMSATS Institute of Information

Technology Islamabad

Ph.D. Mathematics

University of York, UK

Books

• Text Book:

A First Course in Abstract Algebra (7th Edition);

by John B. Fraleigh

• Additional Reading:

Algebra (3rd Edition)

by Serge Lang

Abstract Algebra (1st Edition)

by Robert B. Ash

Grading

• Credit hours (3,0)• Total marks = 100• Sessional 1 = 10 points• Sessional 2 = 15 points• At least 3 quizzes• At least 3 assignments• Final Exam = 50 points

Course Objectives

• Students will be able to write mathematical proofs and reason abstractly in exploring properties of groups and rings

• Use the division algorithm, Euclidean algorithm, and modular arithmetic in computations and proofs about the integers

• Define, construct examples of, and explore properties of groups, including symmetry groups, permutation groups and cyclic groups

Course Objectives cont’d

• Determine subgroups and factor groups of finite groups, determine, use and apply homomorphisms between groups

• Define and construct examples of rings, including integral domains and polynomial rings.

Course Outline

• Groups: Historical background • Definition of a Group with some examples • Order of an element of a group • subgroup, Generators and relations • Free Groups, Cyclic Groups • Finite groups• Group of permutations: Cayley’s Theorem on

permutation groups • Cosets and Lagrange’s theorem

Course Outline Cont’d

• Normal subgroups • Simplicity, Normalizers, Direct Products.• Homomorphism: Factor Groups• Isomorphisms, Automorphism• Isomorphism Theorems• Define and construct examples of rings• Integral domains and polynmial rings.

Chapter 1

Groups and Subgroups

Today’s Topics

• Introduction• Binary operations• Definition of Group

Introduction

Set:

A set is a collection of objects.

Examples:

1. S={1,2,3,…,10}

2. S={The set of all prime numbers upto 10}

3. S={The set of all cities of Pakistan}

4. S={The set of all students of MSc(mathematics) at virtual campus of comsats}

Some very important number sets

• N={1,2,3,…} The set of natural numbers• W={0,1,2,3,…} The set of whole numbers• Z={…,-3,-2,-1,0,1,2,3,…} The set of integers• Q={p/q | p and q are integers with q not equal to

zero} The set of rational numbers• I={The set of irrational numbers}• R={The set of real numbers}=Q U I• C={a+ib| a,b are real numbers} The set of

complex numbers (i= square root (-1))

Subset, proper subset and more definitions:

• Subset • Proper/Improper subset• Empty subset• Union of sets• Intersection of sets

Some basic symbols:

• For all/ For each/ For every • There exist/ There is one• Implies / If then• If and only one /Iff• Such that• Belongs to/ Is in

Function:

Definition

•A function f : A → B between two sets A

(domain) and B (codomain) is a rule that assigns to

each element a in A, a unique element f (a) in B".

•Mathematically f : A → B is a function if

i. f (a) in B, ∀ a in A and

ii. a1 = a2 ⇒ f (a1 ) = f (a2 ), ∀ a1 , a2 in A

Examples

1. Identity function

2. Zero function

3. f:R R such that f(x)= x2 for all x in R

4. f:R R such that f(x)= sqrt(x) for all x in R

Range, 1-1, onto functions

• Let f: A B be a function. The set

f(A)={b in b | f(a)=b for some a in A}

is called the range of f. Note that f(A) is a subset of B and it may or may not equal B.

• A function is called onto if f(A)=B.• A function is called 1-1 if

f(a1)=f(a2) implies a1=a2

• We will call a function bijective is it is both 1-1 and onto.