Group Theory

Main subject categories: • Group theory • Mathematical symmetryThis textbook focuses on the basics and complex themes of group theory taught to senior undergraduate mathematics students across universities. The contents focus on the properties of groups, subgroups, cyclic groups, permutation groups, cosets and Lagrange’s theorem, normal subgroups and factor groups, group homomorphisms and isomorphisms, automorphisms, direct products, group actions and Sylow theorems. Pedagogical elements such as end of chapter exercises and solved problems are included to help understand abstract notions. Intermediate lemmas are also carefully designed so that they not only serve the theorems but are also valuable independently. The book is a useful reference to undergraduate and graduate students besides academics. Preface Foreword Contents List of Symbols 1 Groups 1.1 Groups 1.2 Cayley Table 1.3 Elementary Properties of Groups 1.4 Dihedral Groups 2 Finite Groups and Subgroups 2.1 Finite Groups 2.2 Subgroups 2.3 Subgroup Tests 2.4 Special Class of Subgroups 2.5 Intersection and Union of Subgroups 2.6 Product of Two Subgroups 3 Cyclic Groups 3.1 Cyclic Groups and their Properties 3.2 Generators of a Cyclic Group 3.3 Subgroups of Cyclic Groups 4 Permutation Groups 4.1 Permutation of a Set 4.2 Permutation Group of a Set 4.3 Cycle Notation 4.4 Theorems on Permutations and Cycles 4.5 Even and Odd Permutations 4.6 Alternating Group of Degree n 5 Cosets and Lagrange’s Theorem 5.1 Definition of Cosets and Properties of Cosets 5.2 Lagrange’s Theorem and its Applications 5.3 Application of Cosets to Permutation Groups 6 Normal Subgroups and Factor Groups 6.1 Normal Subgroup and Equivalent Conditions for a Subgroup to be Normal 6.2 Factor Groups 6.3 Commutator Subgroup of a Group and its Properties 6.4 The G/Z Theorem 6.5 Cauchy’s Theorem for Abelian Group 7 Group Homomorphism and Isomorphism 7.1 Homomorphism of Groups and its Properties 7.2 Properties of Subgroups under Homomorphism 7.3 Isomorphism of Groups 7.4 Some Theorems Based on Isomorphism of Groups 8 Automorphisms 8.1 Automorphism of a Group 8.2 Inner Automorphisms 8.3 Theorems Based on Automorphism of a Group 9 Direct Products 9.1 External Direct Product 9.2 Properties of External Direct Products 9.3 U(n) as External Direct Products 9.4 Internal Direct Products 9.5 Fundamental Theorem of Finite Abelian Groups 10 Group Actions 10.1 Group Actions 10.2 Kernels, Orbits and Stabilizers 10.3 Group acting on themselves by Conjugation 10.4 Conjugacy in Sn 11 Sylow Theorems 11.1 p–Groups and Sylow p–Subgroups 11.2 Simple Groups Index