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# Lecture 1 - Basic Definition and Properties of Groups Engineering Mathematics Notes | EduRev

## Engineering Mathematics : Lecture 1 - Basic Definition and Properties of Groups Engineering Mathematics Notes | EduRev

``` 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

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

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

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

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.
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