Lecture 2 - Basic Definition and Properties of Subgroups | Group Theory- Definition, Properties - Engineering Mathematics PDF Download

 Page 1


Basic Definition and Properties of Subgroups 
Institute of Lifelong Learning, University of Delhi                                                 pg. 1 
 
 
 
 
 
 
 
 
 
Lesson: Basic Definition and Properties of Subgroups 
Course Developer: Pragati Gautam and Priyanka Sahni 
Department / College: Assistant Professors, Department 
of Mathematics, Kamala Nehru College 
University of Delhi 
 
 
 
 
 
 
  
Page 2


Basic Definition and Properties of Subgroups 
Institute of Lifelong Learning, University of Delhi                                                 pg. 1 
 
 
 
 
 
 
 
 
 
Lesson: Basic Definition and Properties of Subgroups 
Course Developer: Pragati Gautam and Priyanka Sahni 
Department / College: Assistant Professors, Department 
of Mathematics, Kamala Nehru College 
University of Delhi 
 
 
 
 
 
 
  
Institute of Lifelong Learning, University of Delhi                                                 pg. 2 
 
Table of contents: 
Chapter: Basic Definition and Properties of Subgroups 
? 1 : Learning outcomes  
? 2 : Introduction  
? 3 : Prerequisites and Notations  
? 4: Subgroups  
? 5 : General Properties of Subgroups  
? 6 : Some Basic examples of Subgroups 
? 7: Theorems and Problems on Subgroups 
? 8 : Some Special Subgroups 
? 9 : Cosets and Lagrange?s Theorem 
? Exercises  
? Summary  
? References 
 
  
Page 3


Basic Definition and Properties of Subgroups 
Institute of Lifelong Learning, University of Delhi                                                 pg. 1 
 
 
 
 
 
 
 
 
 
Lesson: Basic Definition and Properties of Subgroups 
Course Developer: Pragati Gautam and Priyanka Sahni 
Department / College: Assistant Professors, Department 
of Mathematics, Kamala Nehru College 
University of Delhi 
 
 
 
 
 
 
  
Institute of Lifelong Learning, University of Delhi                                                 pg. 2 
 
Table of contents: 
Chapter: Basic Definition and Properties of Subgroups 
? 1 : Learning outcomes  
? 2 : Introduction  
? 3 : Prerequisites and Notations  
? 4: Subgroups  
? 5 : General Properties of Subgroups  
? 6 : Some Basic examples of Subgroups 
? 7: Theorems and Problems on Subgroups 
? 8 : Some Special Subgroups 
? 9 : Cosets and Lagrange?s Theorem 
? Exercises  
? Summary  
? References 
 
  
Institute of Lifelong Learning, University of Delhi                                                 pg. 3 
 
1. Learning Outcomes  
After you have read this chapter, you will be able to  
? Define subgroups and construct some examples of Subgroups. 
? Understand the General Properties of subgroups. 
? Check whether a given subset of a group forms its subgroup or not. 
? Construct new Subgroups out of given Subgroups ¸¸using algebra of Subgroups. 
? Understand the concept of Centre, Normalizer and Centralizer and should be able 
to find them for a given group. 
? Define Cosets and understand its importance in a group. 
? State and apply Lagrange?s Theorem in finding the possible orders of subgroups 
(and elements ) in a group.  
  
Page 4


Basic Definition and Properties of Subgroups 
Institute of Lifelong Learning, University of Delhi                                                 pg. 1 
 
 
 
 
 
 
 
 
 
Lesson: Basic Definition and Properties of Subgroups 
Course Developer: Pragati Gautam and Priyanka Sahni 
Department / College: Assistant Professors, Department 
of Mathematics, Kamala Nehru College 
University of Delhi 
 
 
 
 
 
 
  
Institute of Lifelong Learning, University of Delhi                                                 pg. 2 
 
Table of contents: 
Chapter: Basic Definition and Properties of Subgroups 
? 1 : Learning outcomes  
? 2 : Introduction  
? 3 : Prerequisites and Notations  
? 4: Subgroups  
? 5 : General Properties of Subgroups  
? 6 : Some Basic examples of Subgroups 
? 7: Theorems and Problems on Subgroups 
? 8 : Some Special Subgroups 
? 9 : Cosets and Lagrange?s Theorem 
? Exercises  
? Summary  
? References 
 
  
Institute of Lifelong Learning, University of Delhi                                                 pg. 3 
 
1. Learning Outcomes  
After you have read this chapter, you will be able to  
? Define subgroups and construct some examples of Subgroups. 
? Understand the General Properties of subgroups. 
? Check whether a given subset of a group forms its subgroup or not. 
? Construct new Subgroups out of given Subgroups ¸¸using algebra of Subgroups. 
? Understand the concept of Centre, Normalizer and Centralizer and should be able 
to find them for a given group. 
? Define Cosets and understand its importance in a group. 
? State and apply Lagrange?s Theorem in finding the possible orders of subgroups 
(and elements ) in a group.  
  
Institute of Lifelong Learning, University of Delhi                                                 pg. 4 
 
Basic Definition and Properties of Subgroups 
It is tribute to the genius of Galois that he recognized that those subgroups for which the 
left and right cosets coincide are distinguished ones. Very often in Mathematics the 
crucial problem is to recognize and to discover what are the relevant concepts; once this 
is accomplished the job may be more than half done. 
  I.N. Herstein, Topics in Algebra 
2. Introduction: 
 Algebra emerged as a branch of mathematics at the end of the 16
th
 century. 
Algebra was considered similar to arithmetic when computations were done with non-
numerical mathematical objects. The word "algebra" is derived from the Arabic word Al-
Jabr which comes from the treatise written in 820 by the medieval Persian 
mathematician, Muhammad ibn Musa al – Khwarizmi from an Arabic book which in its 
translated form is known as "The compendious Book on Calculation by Completion and 
Balancing". This treatise provided for the systematic solution of linear and quadratic 
equations. 
 The history of algebra began in ancient Egypt and Babylon, where people learned 
to solved linear, quadratic and indeterminate equations whereby several unknowns were 
involved. Diophantus continued the tradition of Egypt and Babylon and wrote a book 
Arithmetica which gives solutions to difficult indeterminate equations. This ancient 
knowledge of solutions of equations in turn found a home early in the Islamic world, 
where it was known as the "Science of restoration and balancing". 
 An important development in algebra in the 16th century was the introduction of 
symbols for the unknown and for algebraic powers and operations. Descartes wrote a 
Book III of La geometrie in 1637 which was like a modern algebra text. By the time of 
Gauss, algebra had entered its modern phase and then the study of abstract 
mathematical systems such as complex numbers came into existence. The widespread 
influence of this abstract approach led George Boole to write "The Laws of Thought" in 
1854 which was based on Logic theory. Since that time, modern algebra also called 
abstract algebra has continued to develop and it has found applications in all branches of 
mathematics and sciences as well. 
 In the present chapter we will be dealing with very vital area of Groups and 
Subgroups in Algebra. Groups will only be introduced whereas properties of Subgroups 
will be discussed in detail. 
Page 5


Basic Definition and Properties of Subgroups 
Institute of Lifelong Learning, University of Delhi                                                 pg. 1 
 
 
 
 
 
 
 
 
 
Lesson: Basic Definition and Properties of Subgroups 
Course Developer: Pragati Gautam and Priyanka Sahni 
Department / College: Assistant Professors, Department 
of Mathematics, Kamala Nehru College 
University of Delhi 
 
 
 
 
 
 
  
Institute of Lifelong Learning, University of Delhi                                                 pg. 2 
 
Table of contents: 
Chapter: Basic Definition and Properties of Subgroups 
? 1 : Learning outcomes  
? 2 : Introduction  
? 3 : Prerequisites and Notations  
? 4: Subgroups  
? 5 : General Properties of Subgroups  
? 6 : Some Basic examples of Subgroups 
? 7: Theorems and Problems on Subgroups 
? 8 : Some Special Subgroups 
? 9 : Cosets and Lagrange?s Theorem 
? Exercises  
? Summary  
? References 
 
  
Institute of Lifelong Learning, University of Delhi                                                 pg. 3 
 
1. Learning Outcomes  
After you have read this chapter, you will be able to  
? Define subgroups and construct some examples of Subgroups. 
? Understand the General Properties of subgroups. 
? Check whether a given subset of a group forms its subgroup or not. 
? Construct new Subgroups out of given Subgroups ¸¸using algebra of Subgroups. 
? Understand the concept of Centre, Normalizer and Centralizer and should be able 
to find them for a given group. 
? Define Cosets and understand its importance in a group. 
? State and apply Lagrange?s Theorem in finding the possible orders of subgroups 
(and elements ) in a group.  
  
Institute of Lifelong Learning, University of Delhi                                                 pg. 4 
 
Basic Definition and Properties of Subgroups 
It is tribute to the genius of Galois that he recognized that those subgroups for which the 
left and right cosets coincide are distinguished ones. Very often in Mathematics the 
crucial problem is to recognize and to discover what are the relevant concepts; once this 
is accomplished the job may be more than half done. 
  I.N. Herstein, Topics in Algebra 
2. Introduction: 
 Algebra emerged as a branch of mathematics at the end of the 16
th
 century. 
Algebra was considered similar to arithmetic when computations were done with non-
numerical mathematical objects. The word "algebra" is derived from the Arabic word Al-
Jabr which comes from the treatise written in 820 by the medieval Persian 
mathematician, Muhammad ibn Musa al – Khwarizmi from an Arabic book which in its 
translated form is known as "The compendious Book on Calculation by Completion and 
Balancing". This treatise provided for the systematic solution of linear and quadratic 
equations. 
 The history of algebra began in ancient Egypt and Babylon, where people learned 
to solved linear, quadratic and indeterminate equations whereby several unknowns were 
involved. Diophantus continued the tradition of Egypt and Babylon and wrote a book 
Arithmetica which gives solutions to difficult indeterminate equations. This ancient 
knowledge of solutions of equations in turn found a home early in the Islamic world, 
where it was known as the "Science of restoration and balancing". 
 An important development in algebra in the 16th century was the introduction of 
symbols for the unknown and for algebraic powers and operations. Descartes wrote a 
Book III of La geometrie in 1637 which was like a modern algebra text. By the time of 
Gauss, algebra had entered its modern phase and then the study of abstract 
mathematical systems such as complex numbers came into existence. The widespread 
influence of this abstract approach led George Boole to write "The Laws of Thought" in 
1854 which was based on Logic theory. Since that time, modern algebra also called 
abstract algebra has continued to develop and it has found applications in all branches of 
mathematics and sciences as well. 
 In the present chapter we will be dealing with very vital area of Groups and 
Subgroups in Algebra. Groups will only be introduced whereas properties of Subgroups 
will be discussed in detail. 
Institute of Lifelong Learning, University of Delhi                                                 pg. 5 
 
 Group theory is the study of Symmetry. It shows up in many areas of geometry, 
classical problems in algebra have been resolved with group theory. The mathematics of 
public-key cryptography uses a lot of group theory. The modern viewpoint of Fourier 
Analysis is a fusion of Analysis, Linear Algebra and Group Theory. On the lighter side, 
there are applications of group theory to puzzles, such as the 15-puzzle and Rubik's 
cube. It provides the conceptual framework for solving such puzzles. 
 Subgroups find a lot of application in Capability analysis. Subgroups should be 
representative of the output from the process you want to evaluate. Subgroups are a 
way to "pre-mix" a number of channels on a sound console before sending them to the 
master output mix (Subgroups is not only a helpful tool while mixing, but is powerful in 
the middle of producing beats as well). Clinicians, when trying to apply trial results to 
patient care, need to individualize patient care and, potentially manage patients based 
on results of subgroup analysis.