The book is intended to serve as an introductory course in group theory geared towards second-year university students. It aims to provide them with the background needed to pursue more advanced courses in algebra and to provide a rich source of examples and exercises. Studying group theory began in the late eighteenth century and is still gaining importance due to its applications in physics, chemistry, geometry, and many fields in mathematics. The text is broadly divided into three parts. The first part establishes the prerequisite knowledge required to study group theory. This includes topics in set theory, geometry, and number theory. Each of the chapters ends with solved and unsolved exercises relating to the topic. By doing this, the authors hope to fill the gaps between all the branches in mathematics that are linked to group theory. The second part is the core of the book which discusses topics on semigroups, groups, symmetric groups, subgroups, homomorphisms, isomorphism, and Abelian groups. The last part of the book introduces SAGE, a mathematical software that is used to solve group theory problems. Here, most of the important commands in SAGE are explained, and many examples and exercises are provided.
Author(s): Bana Al Subaiei, Muneerah Al Nuwairan
Publisher: Springer
Year: 2023
Language: English
Pages: 428
City: Singapore
Preface
Contents
About the Authors
Symbols and Acronyms
List of Figures
List of Tables
1 Background Results in Set Theory
1.1 Operations on Sets
1.2 Principle of Mathematical Induction
1.3 Binary Relations on Sets
1.4 Types of Binary Relations on Sets
1.5 Functions
1.6 Matrices
1.7 Geometric Transformations and Symmetries in the Plane
References
2 Algebraic Operations on Integers
2.1 Basic Algebraic Operations on Integers
2.2 Divisibility of Integers
2.3 Common Divisors of Integers
2.4 Euclidean Algorithm (Euclid’s Algorithm)
2.5 Bézout’s Lemma (Bézout’s Identity)
2.6 Relatively Prime Integers
2.7 Common Multiples of Integers
2.8 Prime Numbers and the Fundamental Theorem of Arithmetic
2.9 Applications of the Fundamental Theorem of Arithmetic
Reference
3 The Integers Modulo n
3.1 Structure of Integers Modulo n
3.2 Functions on the Integers Modulo n
3.3 Algebraic Operations the Integers Modulo n
3.4 The Addition Modulo n and Multiplication Modulo n Tables
3.5 Use of the “mod n” Formula
3.6 Linear Equations on the Integers Modulo n
Reference
4 Semigroups and Monoids
4.1 Binary Operations on Sets
4.2 Semigroups and Monoids
4.3 Invertible Elements in Monoids
4.4 Idempotent Elements in Semigroups
5 Groups
5.1 Definition and Basic Examples
5.2 Cayley’s Tables for Finite Groups
5.3 Additive and Multiplicative Groups of Integers Modulo n
5.4 Abelian Groups and the Center of a Group
5.5 The Order of an Element in a Group
5.6 Direct Product of Groups
Reference
6 The Symmetric Group “An Example of Finite Nonabelian Group”
6.1 Matrix Representation of Permutations
6.2 Cycles on { 1,2, ,n }
6.3 Orbits of a Permutation
6.4 Order of a Permutation
6.5 Odd and Even Permutations
References
7 Subgroups
7.1 Definitions and Basic Examples
7.2 Operations on Subgroups
7.3 Subgroups Generated by a Set and Finitely Generated Subgroups
7.4 Cosets of Subgroups and Lagrange’s Theorem
7.5 Normal Subgroups of a Group
7.6 Internal Direct Product of Subgroups
7.7 The Quotient Groups
References
8 Group Homomorphisms and Isomorphic Groups
8.1 Group Homomorphisms, Definitions, and Basic Examples
8.2 The Kernel and Image of Homomorphism
8.3 Group Isomorphisms and Cayley’s Theorem
8.4 The Fundamental Theorems of Homomorphisms
8.5 Group Actions and Group Homomorphisms
Reference
9 Classification of Finite Abelian Groups
9.1 Cyclic Groups
9.2 Primary Groups
9.3 Independent Subsets, Spanning Subsets, and Bases of a Group
9.4 The Fundamental Theorem of Finite Abelian Groups
References
10 Group Theory and Sage
10.1 What Is Sage?
10.2 Examples for Using Sage in Group Theory
10.2.1 Commands Related to Sets and Basic Operations
10.2.2 Commands Related to Integers Modulo n
10.2.3 Commands Related to Groups
10.2.4 Commands Related to Subgroups
References
Bibliography
Index
