A First Course in Group Theory

Description

This textbook provides a readable account of the examples and fundamental results of groups from a theoretical and geometrical point of view. Topics on important examples of groups (like cyclic groups, permutation groups, group of arithmetical functions, matrix groups and linear groups), Lagrange’s theorem, normal subgroups, factor groups, derived subgroup, homomorphism, isomorphism and automorphism of groups have been discussed in depth. Covering all major topics, this book is targeted to undergraduate students of mathematics with no prerequisite knowledge of the discussed topics. Each section ends with a set of worked-out problems and supplementary exercises to challenge the knowledge and ability of the reader. Preface Contents About the Author Notation 1 Preliminaries Notions 1.1 Sets and Equivalence Relations 1.2 Functions 1.3 Ordered Sets 1.4 Combinatorial Analysis 1.5 Divisibility and Prime Numbers 1.6 Worked-Out Problems 1.7 Supplementary Exercises 2 Symmetries of Shapes 2.1 Symmetry 2.2 Translations 2.3 Rotation Symmetries 2.4 Mirror Reflection Symmetries 2.5 Congruence Transformations 2.6 Worked-Out Problems 2.7 Supplementary Exercises 3 Groups 3.1 A Short History of Group Theory 3.2 Binary Operations 3.3 Semigroups and Monoids (Optional) 3.4 Groups and Examples 3.5 Turning Groups into Latin Squares (Optional) 3.6 Subgroups 3.7 Worked-Out Problems 3.8 Supplementary Exercises 4 Cyclic Groups 4.1 Group of Integers Modulo n 4.2 Cyclic Groups 4.3 Generating Sets 4.4 Worked-Out Problems 4.5 Supplementary Exercises 5 Permutation Groups 5.1 Inverse Functions and Permutations 5.2 Symmetric Groups 5.3 Alternating Groups 5.4 Worked-Out Problems 5.5 Supplementary Exercises 6 Group of Arithmetical Functions (Optional) 6.1 Arithmetical Functions 6.2 Dirichlet Product and Its Properties 6.3 Multiplicative Functions 6.4 Worked-Out Problems 6.5 Supplementary Exercises 7 Matrix Groups 7.1 Introduction to Matrix Groups 7.2 More About Vectors in mathbbRn 7.3 Rotation Groups 7.4 Reflections in mathbbR2 and mathbbR3 7.5 Translation and Scaling Matrices 7.6 Dihedral Groups 7.7 Quaternion Group 7.8 Worked-Out Problems 7.9 Supplementary Exercises 8 Cosets of Subgroups and Lagrange's Theorem 8.1 Cosets and Their Properties 8.2 Geometric Examples of Cosets 8.3 Lagrange's Theorem 8.4 Index of Subgroups 8.5 A Counting Principle and Double Cosets 8.6 Worked-Out Problems 8.7 Supplementary Exercises 9 Normal Subgroups and Factor Groups 9.1 Normal Subgroups 9.2 Factor Groups 9.3 Cauchy's Theorem and Class Equation 9.4 Worked-Out Problems 9.5 Supplementary Exercises 10 Some Special Subgroups 10.1 Commutators and Derived Subgroups 10.2 Derived Subgroups of Some Special Groups 10.3 Maximal Subgroups 10.4 Worked-Out Problems 10.5 Supplementary Exercises 11 Group Homomorphisms 11.1 Homomorphisms and Their Properties 11.2 Isomorphism Theorems 11.3 Cayley's Theorem 11.4 Automorphisms 11.5 Characteristic Subgroups 11.6 Another View of Linear Groups 11.7 Worked-Out Problems 11.8 Supplementary Exercises Appendix References Index