Lecture 1 - Basic Definition and Properties of Groups | Group Theory- Definition, Properties - Engineering Mathematics PDF Download

 Page 1


Basic Definition and Properties of Groups 
Institute of Lifelong Learning, University of Delhi                                                 pg. 1 
 
 
 
 
 
 
 
Subject: Mathematics 
Lesson: Basic Definition and Properties of Groups 
Lesson  Developer: Pragati Gautam 
Department / College: Assistant Professor, Department 
of Mathematics, Kamala Nehru College 
University of Delhi 
 
 
 
 
  
Page 2


Basic Definition and Properties of Groups 
Institute of Lifelong Learning, University of Delhi                                                 pg. 1 
 
 
 
 
 
 
 
Subject: Mathematics 
Lesson: Basic Definition and Properties of Groups 
Lesson  Developer: Pragati Gautam 
Department / College: Assistant Professor, Department 
of Mathematics, Kamala Nehru College 
University of Delhi 
 
 
 
 
  
Basic Definition and Properties of Groups 
Institute of Lifelong Learning, University of Delhi                                                 pg. 2 
 
Table of contents: 
Chapter: Basic Definition and Properties of Groups 
? 1 : Learning outcomes  
? 2 : Introduction  
? 3 : Prerequisites and Notations  
? 4: Groups  
? 5 : General Properties of Groups  
? 6 : Definition of a Group based upon left Axioms  
? 7: Composition Table for Finite sets 
? 8 : Modular Arithmetic 
? 9 : Order of an element of a group. 
? Exercises  
? Summary  
? References 
  
Page 3


Basic Definition and Properties of Groups 
Institute of Lifelong Learning, University of Delhi                                                 pg. 1 
 
 
 
 
 
 
 
Subject: Mathematics 
Lesson: Basic Definition and Properties of Groups 
Lesson  Developer: Pragati Gautam 
Department / College: Assistant Professor, Department 
of Mathematics, Kamala Nehru College 
University of Delhi 
 
 
 
 
  
Basic Definition and Properties of Groups 
Institute of Lifelong Learning, University of Delhi                                                 pg. 2 
 
Table of contents: 
Chapter: Basic Definition and Properties of Groups 
? 1 : Learning outcomes  
? 2 : Introduction  
? 3 : Prerequisites and Notations  
? 4: Groups  
? 5 : General Properties of Groups  
? 6 : Definition of a Group based upon left Axioms  
? 7: Composition Table for Finite sets 
? 8 : Modular Arithmetic 
? 9 : Order of an element of a group. 
? Exercises  
? Summary  
? References 
  
Basic Definition and Properties of Groups 
Institute of Lifelong Learning, University of Delhi                                                 pg. 3 
 
1. Learning outcomes  
After you have read this chapter, you should be able to  
? Define Groups and understand its concept  
? Understand general Properties of groups and relate them to theorems and 
questions  
? Differentiate between sets which form a group and which do not form a group. 
? Form a Composition Table and solve examples based on them  
? Understand the concept of Order of an element of a group. 
  
Page 4


Basic Definition and Properties of Groups 
Institute of Lifelong Learning, University of Delhi                                                 pg. 1 
 
 
 
 
 
 
 
Subject: Mathematics 
Lesson: Basic Definition and Properties of Groups 
Lesson  Developer: Pragati Gautam 
Department / College: Assistant Professor, Department 
of Mathematics, Kamala Nehru College 
University of Delhi 
 
 
 
 
  
Basic Definition and Properties of Groups 
Institute of Lifelong Learning, University of Delhi                                                 pg. 2 
 
Table of contents: 
Chapter: Basic Definition and Properties of Groups 
? 1 : Learning outcomes  
? 2 : Introduction  
? 3 : Prerequisites and Notations  
? 4: Groups  
? 5 : General Properties of Groups  
? 6 : Definition of a Group based upon left Axioms  
? 7: Composition Table for Finite sets 
? 8 : Modular Arithmetic 
? 9 : Order of an element of a group. 
? Exercises  
? Summary  
? References 
  
Basic Definition and Properties of Groups 
Institute of Lifelong Learning, University of Delhi                                                 pg. 3 
 
1. Learning outcomes  
After you have read this chapter, you should be able to  
? Define Groups and understand its concept  
? Understand general Properties of groups and relate them to theorems and 
questions  
? Differentiate between sets which form a group and which do not form a group. 
? Form a Composition Table and solve examples based on them  
? Understand the concept of Order of an element of a group. 
  
Basic Definition and Properties of Groups 
Institute of Lifelong Learning, University of Delhi                                                 pg. 4 
 
It [group theory] provides a sensitive instrument for investigating 
symmetry, one of the most pervasive and elemental phenomena of the 
real world. 
M.I. Kargapolov and Ju.I. Marzljakov, 
Fundamentals of the theory of groups. 
 
2. Introduction: 
 Towards the end of the 16
th
 Century, Algebra emerged as a branch of 
Mathematics. Francois Viete was associated with this work. The word "algebra" is 
derived from the Arabic word Al-Jabr and this comes from the treatise written in 
Baghdad in about 825 A.D by the medieval Persian Mathematician, Mohammed ibn-Musa 
al-khowarizmi in his book "Hidab al-jabr  wal-muqubala". The words jabr (JAH-ber) and 
muqubala (moo-KAH-ba-lah.) were used by al-khowarizmi to designate two basic 
operations in solving equations. Jabr was used to transpose subtracted terms to the 
other side of the equation where as muqubalah was to cancel like terms on opposite 
sides of the equation. 
 The origin of algebra can also be traced to the ancient Babylonians who 
developed a positional number system which helped them in solving their rhetorical 
algebraic equations. The Babylonians were always interested in approximate solutions so 
they used linear interpolation to approximate intermediate values. 
 Algebra is a very unique discipline. It is abstract and it is this abstractness of the 
subject that causes the brain to think in totally new patterns. The thinking process 
sharpens the working of brain resulting in a better performance. Once the brain is 
stimulated to think it can do more complex things as the dendrites of the brain grow 
more complex and make good connections with other brain cells. As it is rightly said by 
someone," The study of algebra helps in building more highways upon which future 
cargo can be transported." 
 Algebra is the essential language of mathematics. It deals with two ideas namely 
Variables and functions. Variables are symbols that can represent not only a number but 
also a changing quantity whereas a function is a well defined relationship between two 
variables in which change in one value causes the change in other value. The concept of 
variables and functions help us to define the physical laws that govern our universe and 
help us to understand how our world works. 
 In the present chapter we will be dealing with very vital area of Groups and Sub-
groups in Algebra. Groups are of great interest for mathematicians because they are 
Page 5


Basic Definition and Properties of Groups 
Institute of Lifelong Learning, University of Delhi                                                 pg. 1 
 
 
 
 
 
 
 
Subject: Mathematics 
Lesson: Basic Definition and Properties of Groups 
Lesson  Developer: Pragati Gautam 
Department / College: Assistant Professor, Department 
of Mathematics, Kamala Nehru College 
University of Delhi 
 
 
 
 
  
Basic Definition and Properties of Groups 
Institute of Lifelong Learning, University of Delhi                                                 pg. 2 
 
Table of contents: 
Chapter: Basic Definition and Properties of Groups 
? 1 : Learning outcomes  
? 2 : Introduction  
? 3 : Prerequisites and Notations  
? 4: Groups  
? 5 : General Properties of Groups  
? 6 : Definition of a Group based upon left Axioms  
? 7: Composition Table for Finite sets 
? 8 : Modular Arithmetic 
? 9 : Order of an element of a group. 
? Exercises  
? Summary  
? References 
  
Basic Definition and Properties of Groups 
Institute of Lifelong Learning, University of Delhi                                                 pg. 3 
 
1. Learning outcomes  
After you have read this chapter, you should be able to  
? Define Groups and understand its concept  
? Understand general Properties of groups and relate them to theorems and 
questions  
? Differentiate between sets which form a group and which do not form a group. 
? Form a Composition Table and solve examples based on them  
? Understand the concept of Order of an element of a group. 
  
Basic Definition and Properties of Groups 
Institute of Lifelong Learning, University of Delhi                                                 pg. 4 
 
It [group theory] provides a sensitive instrument for investigating 
symmetry, one of the most pervasive and elemental phenomena of the 
real world. 
M.I. Kargapolov and Ju.I. Marzljakov, 
Fundamentals of the theory of groups. 
 
2. Introduction: 
 Towards the end of the 16
th
 Century, Algebra emerged as a branch of 
Mathematics. Francois Viete was associated with this work. The word "algebra" is 
derived from the Arabic word Al-Jabr and this comes from the treatise written in 
Baghdad in about 825 A.D by the medieval Persian Mathematician, Mohammed ibn-Musa 
al-khowarizmi in his book "Hidab al-jabr  wal-muqubala". The words jabr (JAH-ber) and 
muqubala (moo-KAH-ba-lah.) were used by al-khowarizmi to designate two basic 
operations in solving equations. Jabr was used to transpose subtracted terms to the 
other side of the equation where as muqubalah was to cancel like terms on opposite 
sides of the equation. 
 The origin of algebra can also be traced to the ancient Babylonians who 
developed a positional number system which helped them in solving their rhetorical 
algebraic equations. The Babylonians were always interested in approximate solutions so 
they used linear interpolation to approximate intermediate values. 
 Algebra is a very unique discipline. It is abstract and it is this abstractness of the 
subject that causes the brain to think in totally new patterns. The thinking process 
sharpens the working of brain resulting in a better performance. Once the brain is 
stimulated to think it can do more complex things as the dendrites of the brain grow 
more complex and make good connections with other brain cells. As it is rightly said by 
someone," The study of algebra helps in building more highways upon which future 
cargo can be transported." 
 Algebra is the essential language of mathematics. It deals with two ideas namely 
Variables and functions. Variables are symbols that can represent not only a number but 
also a changing quantity whereas a function is a well defined relationship between two 
variables in which change in one value causes the change in other value. The concept of 
variables and functions help us to define the physical laws that govern our universe and 
help us to understand how our world works. 
 In the present chapter we will be dealing with very vital area of Groups and Sub-
groups in Algebra. Groups are of great interest for mathematicians because they are 
Basic Definition and Properties of Groups 
Institute of Lifelong Learning, University of Delhi                                                 pg. 5 
 
widely used in several branches of mathematics and they also possess a rich theory and 
unify several different contexts.  
Groups are used to classify symmetrical objects. It is of great use in 
Crystallography and in study of molecular structures. Group theory is the mathematics of 
symmetry ? a fundamental notion in science, maths and engineering. There are many 
important practical applications of modular arithmetic that are best understood by 
viewing the modular arithmetic in a group theory framework. Examples include the 
check digits on UPC codes on retail items, ISBN number on books and credit card 
numbers. The Hamming code used in communication systems for automatic error 
correction are groups.  
 The present chapter illustrates the importance of Groups and Sub-groups as one 
of the most classified topics in Algebra. We will discuss various properties of Groups and 
Subgroups and examples related to them.  
 
3.  Prerequisites and Notations 
 Before we formally define a group we should know that group is a system 
consisting of a non-void set G and binary composition on G satisfying some postulates. 
We will discuss some basic definitions and concepts without going  into depth as they are 
in fact the building blocks for the development of the fascinating theory of groups. 
3.1. Definition : Sets : A  collection of well defined objects is called a set. The concept 
of set is most basic in mathematics. Almost all mathematical systems are certain 
collection of sets. 
3.2. Definition : Binary Relation: Let X and Y be two non- empty sets. Then any 
subset of X ? Y is called a binary relation of X to Y. It is also called a "Correspondence”. 
 If X = Y = S, then a subset of  S ?S is called a Binary relation on S. 
3.3 Definition : Mappings : Let X and Y be two non-empty sets. A subset  T of X ?Y is 
called a mapping of X into Y if for each x ? ? X, there exists one and only one y ? Y such 
that (x, y) ? T. A mapping is also known as a function.  
 If (x, y) ? T then y is called the value of T at x for any x ? X or y is said to 
correspond to x under T and we write y = T(x). If T describes the mapping  of X into Y, 
then X is called the domain of T. The set {y ?Y | y = T (x) for x ? X} is called the range 
of T (range T) and Y is called co-domain of T.