A gentle introduction to group theory

A gentle introduction to group theory / Bana Al Subaiei, Muneerah Al Nuwairan.

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...

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Bibliographic Details
Main Author: Al Subaiei, Bana
Other Authors: Al Nuwairan, Muneerah
Format: eBook
Language:English
Published: Singapore : Springer, 2023.
Subjects:
Group theory.
Group theory
Teoria de grups.
Llibres electrònics.
Online Access:Click for online access
Table of Contents:
  • Intro
  • 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

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