ABOUT THE COURSE:This course introduces abstract groups, starting with binary operations on a set and moving through definitions and basic properties. We will explore cyclic groups, symmetries of a triangle and square, direct products, normal subgroups, quotient groups, homomorphisms, kernels, and isomorphisms. Key theorems covered include Lagrange's theorem, Cauchy's theorem, and the three isomorphism theorems. Emphasis is placed on worked-out examples and problem-solving, with weekly lectures featuring explicit calculations to enhance understanding and practical skills. By the end of the course, students will grasp fundamental group structures, apply major theorems, and solve complex problems using group theory principles. This course is ideal for those looking to deepen their understanding of abstract algebra.INTENDED AUDIENCE: Students/research scholar and tearcherPREREQUISITES: Basic of Set Theory and Number Theory
Overview
Syllabus
Week 1: Binary Operations, Introduction to Groups, Examples to Groups, Elementary properties of GroupsWeek 2:Subgroups and some basic properties, Product of two subgroups, Cyclic groups, Generators, its examples and related resultsWeek 3:Classification of Subgroups of Cyclic Groups, Problems on subgroups of cyclic group, Order of an elementWeek 4:Permutation Groups, Symmetric Group (Sn)Week 5:Even and Odd Permutation, Permutation as product Transpositions, examples and problemsWeek 6:Alternating Groups, The Dihedral Group Cosets and their properties, examplesWeek 7:Lagrange's theorem and its applications, Index of a subgroup of a group, Euler's theorem, Fermat's theorem and problemsWeek 8:Normal Subgroups and its some basic properties, Quotient groups with examples and problemsWeek 9:Group Homomorphisms and examples, Properties of Homomorphisms and its examplesWeek 10:Isomorphism, Properties of Isomorphism The Isomorphism TheoremsWeek 11:Automorphism, Cayley's Theorem and its examplesWeek 12:Internal and external direct product of groups and their related results, Structure Theorem of Finite Abelian groups
Taught by
Prof. Nadeem ur Rehman
