INTRODUCTION SC116: Algebraic Structures Short Title of the Course: ALG Instructor: Professor Samaresh Chatterji

ALG Credit Structure: 3 – 1 – 0 – 4 (i.e. 3 lectures + 1 tutorial + 0 practical per week, giving a total of 4 credits for the course) Timetable: –3 Lectures per Week - Tu, W, F 08:30 –1 Tutorial per Week - in Groups –Tutors – Prof V P Sinha and Prof V Sunitha

EVALUATION Tutorials & Quizzes - 15% Insem Examinations (two) - 35% Endsem Examination - 50% Total marks out of 100 will be converted to a letter grade using a curve Above may be slightly modified later with due notice

OBJECTIVE This course helps students understand algebraic structures as underlying specific objects, computations, and systems, develops familiarity with the key algebraic structures which are most frequently encountered: groups, rings, fields, vector spaces, both abstractly in terms of axioms and concretely in terms of the most important examples. It also makes them acquainted with the concept of homomorphisms of algebraic structures in general and in its specific manifestation in the context of the different examples. It also provides knowledge of specific applications of the above understanding, both in attacking mathematical problems and in ICT.

CONTENTS Groups (Subgroups, Isomorphism and Homomorphism, Cosets, Product of Groups, Quotient Groups) Vector Spaces (Fields, Vector Spaces, Subspaces, Bases and Dimension, Co-ordinates), Linear Transformations (Algebra of Linear Transformations, Isomorphism, Matrix Representations), Linear Equations (System of Linear Equations, Elementary Row Operations, RREF, Invertible Matrices), Linear Functionals (including the double dual and the transpose), Eigenvalues and Eigenvectors (Characteristic Polynomial, Orthogonal and Unitary Matrices, Diagonalisation, Systems of Differential Equations, Matrix Exponential) Polynomials (Algebra of Polynomials, Irreducible Polynomials, Prime Factorization).

BOOKS Artin: Algebra, 7th ed, Prentice-Hall, 2001 Datta: Matrix and Linear Algebra, Prentice Hall Herstein: Topics in Algebra, Indian edition, John Wiley Hoffman and Kunze: Linear Algebra, 2nd ed, Prentice-Hall Lay: Linear Algebra and Its Applications, 2nd ed, Addison-Wesley Scheinerman: Mathematics: A Discrete Introduction, Brooks /Cole (This list may be modified later)

Standard Notations The set of natural numbers consists of zero and the positive whole numbers: we shall use the notation: N = {0,1,2,3,………} The set of integers consists of zero and the positive and negative whole numbers: we shall use the notation: Z = {…..,–3, –2, –1,0,1,2,3,………} The set of rational numbers contains all fractions of the form a/b where a and b are integers and b  0. Two rational numbers a/b and c/d are equal exactly when ad = bc. We shall denote the set of rational numbers by Q. We shall also refer to the set of real numbers R and the set of complex numbers C.

Operations on Sets Basic Idea: An algebraic structure is a set together with one or more operations on the set; to make the idea more productive, the operation(s) are required to be well- behaved, i.e. to possess one or more properties. Definition: Let A be a set. An operation (sometimes called a binary operation) on A is a function whose domain contains A  A. Notation: However, we usually use a different notation for operations. Instead of writing f((a,b)) to indicate the image we place a suitable symbol between a and b. Commonly used symbols are: , , , , , etc. Symbols such as: +, –,  are also used, but usually restricted to their standard usages.

Properties of Operations Properties of Operations on a Set: Let  be an operation on a set A. 1.  is said to be closed on A provided a  b  A for all a, b  A. 2.  is said to be associative on A provided (a  b)  c = a  (b  c) for all a, b, c  A. 3.  is said to be commutative on A provided a  b = b  a for all a, b  A. 4.An element e  A is said to be an identity element (or neutral element) for  provided a  e = e  a = a for all a  A. Remark: There can be at most one identity element for an operation  on a set A. 5.Suppose that the operation  on the set A has an identity element e, and suppose that a  A. An element b is said to be an inverse of a provided a  b = b  a = e.

Groups Definition: Let  be an operation on a set G. We call the pair (G,  ) a group provided: 1.The operation  is closed on G, that is, g  h  G for all g, h  G. 2.The operation  is associative, that is, (g  h)  k = g  (h  k) for all g, h, k  G. 3.There is an identity element e  G such that g  e = e  g = g for all g  G. 4.For every element g  G, there is an inverse element h  G, such that g  h = h  g = e. Notation: To have a group, we need to have both a set G and an operation on G. We can use any suitable symbol for the operation; if it is a previously known operation, we may use the standard symbol. If we are talking about general groups, we will use either  or no symbol at all. Definition: Let (G,  ) be a group. If the operation  is also commutative, the group will be called an abelian group. Groups which are not abelian are sometimes referred to as nonabelian.