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.Read More

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