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 (7 th Edition); by John B. Fraleigh Additional Reading: Algebra (3 rd Edition) by Serge Lang Abstract Algebra (1 st 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. a 1 = a 2 ⇒ f (a 1 ) = f (a 2 ), ∀ a 1, a 2 in A

Examples 1.Identity function 2.Zero function 3.f:R R such that f(x)= x 2 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(a 1 )=f(a 2 ) implies a 1 =a 2 We will call a function bijective is it is both 1-1 and onto.