This document provides an overview of abstract algebra and key concepts such as groups. It discusses how the word "algebra" is derived from an Arabian word meaning "union of broken parts." It also outlines important mathematicians who contributed to the development of algebra, such as Al Khwarizmi, the "father of algebra." The document defines what a set and group are, including the properties a group must satisfy like closure, associative, identity, and inverse elements. Examples of groups are given like integers, rational numbers, and matrices. Applications of group theory in fields like physics, chemistry, and technology are mentioned.
This document discusses cyclic groups and their properties. It begins by defining a cyclic group as a group that can be generated by one of its elements. It then provides examples of cyclic groups like the integers under addition and groups of integers modulo n. The key properties of cyclic groups are then outlined, including that cyclic groups are abelian, and the criteria for determining subgroup order and generators. Finite cyclic groups are shown to have unique subgroups for each divisor of the group order. The document concludes by discussing the classification and enumeration of subgroups in cyclic groups.
This document provides a summary and review of key concepts from Abstract Algebra Part 1, including:
- Groups and their properties of closure, associativity, identity, and inverses
- Examples of finite and infinite, abelian and non-abelian groups
- Subgroups, including cyclic subgroups, tests for subgroups, and examples
- Additional concepts like the order of an element, conjugation, and the center of a group
The document defines and discusses several key concepts relating to groups in abstract algebra:
- A group is defined as a non-empty set together with a binary operation that satisfies closure, associativity, identity, and inverse properties.
- An abelian (or commutative) group is one where the binary operation is commutative. Examples of abelian groups include integers under addition.
- The quaternion group is a non-abelian group of order 8 under multiplication.
- Theorems are presented regarding the uniqueness of identity and inverses in a group, as well as cancellation, reverse order, and inverse properties of groups.
The document provides an overview of linear algebra and matrix theory. It discusses the history and development of matrices, defines key matrix concepts like dimensions and operations, and covers foundational topics like matrix addition, multiplication, inverses, and solving systems of linear equations. The document is intended as an introduction to linear algebra and matrices for students.
This document defines cyclic groups and discusses their order theorem. It explains that a cyclic group G is generated by a single element a, such that every element of G can be expressed as an power of a. It then proves that the order of a cyclic group equals the order of its generator using the division algorithm, showing that for any integer m, m can be expressed as nq + r, where n is the order of the generator and 0 <= r < n. It provides examples of applications of group theory in fields like physics, biology, crystal structure analysis, and coding theory.
This document contains a lesson plan for teaching polynomial functions in mathematics to 10th grade students. It includes opening prayers and attendance, a review of concepts, physical activities to reinforce concepts, examples worked out in groups, and individual assessments. The goal is for students to understand how to write polynomial functions in standard form and identify the degree, leading coefficient, and constant term. Students participate in group work and presentations, are provided feedback, and have a post-assessment to check understanding before being assigned practice on graphing calculators.
This document defines and explains key concepts in sets and set theory, including:
- Defining sets and listing elements using roster and set-builder notation
- Important mathematical sets like natural numbers, integers, and real numbers
- Describing relationships between sets such as subsets, equality, and Venn diagrams
- Calculating cardinality to determine the number of elements in a set
- Forming Cartesian products to combine elements from multiple sets into ordered pairs
This document provides an overview of abstract algebra and key concepts such as groups. It discusses how the word "algebra" is derived from an Arabian word meaning "union of broken parts." It also outlines important mathematicians who contributed to the development of algebra, such as Al Khwarizmi, the "father of algebra." The document defines what a set and group are, including the properties a group must satisfy like closure, associative, identity, and inverse elements. Examples of groups are given like integers, rational numbers, and matrices. Applications of group theory in fields like physics, chemistry, and technology are mentioned.
This document discusses cyclic groups and their properties. It begins by defining a cyclic group as a group that can be generated by one of its elements. It then provides examples of cyclic groups like the integers under addition and groups of integers modulo n. The key properties of cyclic groups are then outlined, including that cyclic groups are abelian, and the criteria for determining subgroup order and generators. Finite cyclic groups are shown to have unique subgroups for each divisor of the group order. The document concludes by discussing the classification and enumeration of subgroups in cyclic groups.
This document provides a summary and review of key concepts from Abstract Algebra Part 1, including:
- Groups and their properties of closure, associativity, identity, and inverses
- Examples of finite and infinite, abelian and non-abelian groups
- Subgroups, including cyclic subgroups, tests for subgroups, and examples
- Additional concepts like the order of an element, conjugation, and the center of a group
The document defines and discusses several key concepts relating to groups in abstract algebra:
- A group is defined as a non-empty set together with a binary operation that satisfies closure, associativity, identity, and inverse properties.
- An abelian (or commutative) group is one where the binary operation is commutative. Examples of abelian groups include integers under addition.
- The quaternion group is a non-abelian group of order 8 under multiplication.
- Theorems are presented regarding the uniqueness of identity and inverses in a group, as well as cancellation, reverse order, and inverse properties of groups.
The document provides an overview of linear algebra and matrix theory. It discusses the history and development of matrices, defines key matrix concepts like dimensions and operations, and covers foundational topics like matrix addition, multiplication, inverses, and solving systems of linear equations. The document is intended as an introduction to linear algebra and matrices for students.
This document defines cyclic groups and discusses their order theorem. It explains that a cyclic group G is generated by a single element a, such that every element of G can be expressed as an power of a. It then proves that the order of a cyclic group equals the order of its generator using the division algorithm, showing that for any integer m, m can be expressed as nq + r, where n is the order of the generator and 0 <= r < n. It provides examples of applications of group theory in fields like physics, biology, crystal structure analysis, and coding theory.
This document contains a lesson plan for teaching polynomial functions in mathematics to 10th grade students. It includes opening prayers and attendance, a review of concepts, physical activities to reinforce concepts, examples worked out in groups, and individual assessments. The goal is for students to understand how to write polynomial functions in standard form and identify the degree, leading coefficient, and constant term. Students participate in group work and presentations, are provided feedback, and have a post-assessment to check understanding before being assigned practice on graphing calculators.
This document defines and explains key concepts in sets and set theory, including:
- Defining sets and listing elements using roster and set-builder notation
- Important mathematical sets like natural numbers, integers, and real numbers
- Describing relationships between sets such as subsets, equality, and Venn diagrams
- Calculating cardinality to determine the number of elements in a set
- Forming Cartesian products to combine elements from multiple sets into ordered pairs
This document contains a lesson on slope of a line from a mathematics course. It includes examples of calculating slope given two points on a line, identifying whether graphs represent constant or variable rates of change, and word problems applying slope to real-world contexts like cost of fruit and gas. The key points are that slope is defined as the ratio of rise over run, or change in y over change in x, and represents the constant rate of change for linear equations and functions.
This document provides an overview of matrices and matrix operations. It defines what a matrix is and discusses matrix order and elements. It then covers basic matrix operations like scalar multiplication, addition, and multiplication. It introduces the concepts of transpose, special matrices like diagonal and triangular matrices, and the null and identity matrices. The document aims to define fundamental matrix concepts and arithmetic operations.
Abstract algebra & its applications (1)drselvarani
This document provides information about a state level workshop on abstract algebra and its applications that was held on August 28, 2015 at Sri Sarada Niketan College for Women in Amaravathipudur, India. The workshop included a presentation by Dr. S. SelvaRani, the principal of the college, on the topic of abstract algebra and its applications. Abstract algebra is the study of algebraic structures like groups, rings, and fields. It has many applications in areas like number theory, geometry, physics, and more. Representation theory is also discussed as an important branch of abstract algebra.
Detailed Lesson plan of Product Rule for Exponent Using the Deductive MethodLorie Jane Letada
The document outlines the procedures for a lesson on the product rule for exponent-like terms with exponents. It includes the objectives, subject content, materials, and steps of the lesson. The teacher leads the students in examples of applying the product rule to simplify expressions with the same bases and adds the exponents. Students then practice applying the rule to example expressions on their own.
1. The document discusses groups, subgroups, cosets, normal subgroups, quotient groups, and homomorphisms.
2. It defines cosets, proves Lagrange's theorem that the order of a subgroup divides the order of the group, and provides examples of finding cosets.
3. Normal subgroups are introduced, and it is shown that the set of cosets of a normal subgroup forms a group under a defined operation, known as the quotient group. Homomorphisms between groups are defined, and examples are given.
This document provides an algebra lesson plan on the multiplication and division properties of exponents. The lesson introduces students to exploring exponential functions graphically using Desmos. By graphing expressions with like bases that are multiplied or divided, students observe that the exponents are added for multiplication and subtracted for division. Through this activity, students generalize the multiplication property as "when multiplying expressions with the same base, add the exponents of each base" and the division property as "when dividing expressions with the same base, subtract the exponents of each base." Students then practice applying these properties to simplify expressions without a graphing calculator.
This document discusses geometric sequences and provides examples. It defines a geometric sequence as a sequence where each term is found by multiplying or dividing the same value from one term to the next. An example given is 2, 4, 8, 16, 32, 64, 128, etc. The document also provides the general formula for a geometric sequence as {a, ar, ar2, ar3, ...} where a is the first term and r is the common ratio between terms. It gives practice problems for finding missing terms and the common ratio of geometric sequences.
1. The document introduces groups and related concepts in mathematics.
2. It defines a group as a set with a binary operation that satisfies associativity, identity, and inverse properties. Abelian groups are groups where the binary operation is commutative.
3. Examples of groups include the complex numbers under multiplication, rational numbers under addition, and translations of the plane under composition. Subgroups are subsets of a group that are also groups under the same binary operation.
This document discusses how to solve radical equations by isolating the radical expression, removing the radical sign by raising both sides to the appropriate power, and solving the resulting equation. It provides examples of solving various radical equations step-by-step and checking solutions. Key steps include isolating the radical term, removing the radical, solving the resulting equation, and checking for extraneous roots.
This document discusses the inverse of matrices. It defines the cofactor method for finding the inverse of a matrix, which involves calculating the matrix of cofactors and then taking its transpose divided by the determinant of the original matrix. Several examples are worked through, including calculating the inverse of a 3x3 matrix. The document also discusses using matrices to represent and solve systems of simultaneous linear equations, developing the general matrix solution of x = A^-1b where A is the coefficient matrix, x is the vector of unknowns, and b is the vector of constants.
This document explains the distance formula and how to use it to calculate the distance between points on a Cartesian plane. It provides the steps to use the Pythagorean theorem to derive the distance formula: take the difference between the x-coordinates squared and add it to the difference between the y-coordinates squared, and take the square root of the result. It then works through an example of using the distance formula to calculate the distance between points (3,2) and (8,7), which equals 5 units. The document concludes with practice problems and assignments applying the distance formula.
Group Theory and Its Application: Beamer Presentation (PPT)SIRAJAHMAD36
This document provides an overview of a seminar presentation on group theory and its applications. The presentation covers topics such as the definition of groups, order of groups and group elements, modular arithmetic, subgroups, Lagrange's theorem, and Sylow's theorems. It also discusses some examples of groups and applications of group theory in fields like algebraic topology, number theory, and physics. The presentation aims to introduce fundamental concepts in modern algebra through group theory.
This document contains a semi-detailed lesson plan for teaching dividing rational expressions in Math 8. The lesson plan outlines intended learning outcomes, learning content including the subject matter and references, required materials, and learning experiences such as sample activities, analysis questions, and applications. It provides examples of dividing rational expressions and asks students to divide additional rational expressions as an evaluation of their understanding.
Factorials are the product of all positive integers from a given number down to 1. This document provides examples of calculating factorials such as 4! = 24 and discusses the rules for operations involving factorials such as cancellation rules that apply when dividing factorials. Practice problems are included with explanations to reinforce understanding of how to calculate and operate with factorial expressions.
This document discusses different methods for solving linear equation systems, including LU factorization methods like Doolittle, Crout, and Cholesky decomposition. It provides examples of applying each method to solve systems of equations step-by-step. The Doolittle method assumes the diagonal of the lower triangular matrix L has ones. The Crout method assumes the diagonal of the upper triangular matrix U has ones. The Cholesky method decomposes the matrix into the product of a lower triangular matrix and its transpose. It explains that LU decomposition costs 2n^3/3 flops to decompose the matrix and solve the system, making it faster than directly inverting the matrix.
This document discusses group theory concepts including:
1) Definitions of groups, abelian groups, order of groups and elements.
2) Properties of cyclic groups, including examples like Zn and Z.
3) Introduction to normal subgroups and their properties. Factor groups and homomorphisms are also discussed.
The document defines relations and functions. A relation is a set of ordered pairs, while a function is a special type of relation where each x-value is mapped to only one y-value. The domain is the set of x-values and the range is the set of y-values. Functions can be identified using the vertical line test or by mapping the relation to check if any x-values are mapped to multiple y-values. Evaluating functions involves substituting domain values into the function rule to find the corresponding range values.
This document discusses two types of combinations: combinations with repetition and combinations without repetition. Combinations with repetition allow elements to be chosen multiple times from a set, while combinations without repetition do not allow repeats. It provides examples to illustrate each type, such as how many 2-letter words can be created from 4 letters with repetition (10 combinations) versus without repetition (6 combinations). The key difference between the two types is whether repetition of elements is allowed or not when selecting elements from a set.
Permutations involve arranging objects in a particular order, where order matters. There are a few different formulas to calculate the number of permutations depending on the situation. The basic formula for permutations of n objects taken r at a time is nPr = n!/(n-r)!. Some examples provided calculate the number of permutations when arranging letters, skiers in a competition, and people entering a cave.
The document discusses the standard form of a circle equation and provides examples of writing the equation of various circles given their center and radius. It begins by defining the components of a circle - the center (h,k) and radius r. Then it shows the standard form of a circle equation: (x-h)2 + (y-k)2 = r2. The rest of the document provides the center and radius of several circles and has the reader write out the corresponding standard equation for each one.
The document defines what a group is in mathematics. A group is a set with an operation that is associative, has a neutral element, and where each element has an inverse. Some examples of groups are the integers under addition, rational numbers under addition, and non-zero real numbers under multiplication. Finite groups with a set number of elements, like integers modulo n, are especially important for scientific applications. Not all groups are commutative, as shown by the group of matrices under multiplication.
This document provides an overview of radicals and their properties. It discusses:
- Simplifying radicals using product, quotient, and nth root rules
- Adding and subtracting radicals by combining like terms
- Multiplying and dividing radicals using distribution and rationalizing denominators
- Solving equations containing radicals by isolating radicals and raising both sides to matching powers
- Applications including using the Pythagorean theorem and distance formula to solve problems involving radicals
This document contains a lesson on slope of a line from a mathematics course. It includes examples of calculating slope given two points on a line, identifying whether graphs represent constant or variable rates of change, and word problems applying slope to real-world contexts like cost of fruit and gas. The key points are that slope is defined as the ratio of rise over run, or change in y over change in x, and represents the constant rate of change for linear equations and functions.
This document provides an overview of matrices and matrix operations. It defines what a matrix is and discusses matrix order and elements. It then covers basic matrix operations like scalar multiplication, addition, and multiplication. It introduces the concepts of transpose, special matrices like diagonal and triangular matrices, and the null and identity matrices. The document aims to define fundamental matrix concepts and arithmetic operations.
Abstract algebra & its applications (1)drselvarani
This document provides information about a state level workshop on abstract algebra and its applications that was held on August 28, 2015 at Sri Sarada Niketan College for Women in Amaravathipudur, India. The workshop included a presentation by Dr. S. SelvaRani, the principal of the college, on the topic of abstract algebra and its applications. Abstract algebra is the study of algebraic structures like groups, rings, and fields. It has many applications in areas like number theory, geometry, physics, and more. Representation theory is also discussed as an important branch of abstract algebra.
Detailed Lesson plan of Product Rule for Exponent Using the Deductive MethodLorie Jane Letada
The document outlines the procedures for a lesson on the product rule for exponent-like terms with exponents. It includes the objectives, subject content, materials, and steps of the lesson. The teacher leads the students in examples of applying the product rule to simplify expressions with the same bases and adds the exponents. Students then practice applying the rule to example expressions on their own.
1. The document discusses groups, subgroups, cosets, normal subgroups, quotient groups, and homomorphisms.
2. It defines cosets, proves Lagrange's theorem that the order of a subgroup divides the order of the group, and provides examples of finding cosets.
3. Normal subgroups are introduced, and it is shown that the set of cosets of a normal subgroup forms a group under a defined operation, known as the quotient group. Homomorphisms between groups are defined, and examples are given.
This document provides an algebra lesson plan on the multiplication and division properties of exponents. The lesson introduces students to exploring exponential functions graphically using Desmos. By graphing expressions with like bases that are multiplied or divided, students observe that the exponents are added for multiplication and subtracted for division. Through this activity, students generalize the multiplication property as "when multiplying expressions with the same base, add the exponents of each base" and the division property as "when dividing expressions with the same base, subtract the exponents of each base." Students then practice applying these properties to simplify expressions without a graphing calculator.
This document discusses geometric sequences and provides examples. It defines a geometric sequence as a sequence where each term is found by multiplying or dividing the same value from one term to the next. An example given is 2, 4, 8, 16, 32, 64, 128, etc. The document also provides the general formula for a geometric sequence as {a, ar, ar2, ar3, ...} where a is the first term and r is the common ratio between terms. It gives practice problems for finding missing terms and the common ratio of geometric sequences.
1. The document introduces groups and related concepts in mathematics.
2. It defines a group as a set with a binary operation that satisfies associativity, identity, and inverse properties. Abelian groups are groups where the binary operation is commutative.
3. Examples of groups include the complex numbers under multiplication, rational numbers under addition, and translations of the plane under composition. Subgroups are subsets of a group that are also groups under the same binary operation.
This document discusses how to solve radical equations by isolating the radical expression, removing the radical sign by raising both sides to the appropriate power, and solving the resulting equation. It provides examples of solving various radical equations step-by-step and checking solutions. Key steps include isolating the radical term, removing the radical, solving the resulting equation, and checking for extraneous roots.
This document discusses the inverse of matrices. It defines the cofactor method for finding the inverse of a matrix, which involves calculating the matrix of cofactors and then taking its transpose divided by the determinant of the original matrix. Several examples are worked through, including calculating the inverse of a 3x3 matrix. The document also discusses using matrices to represent and solve systems of simultaneous linear equations, developing the general matrix solution of x = A^-1b where A is the coefficient matrix, x is the vector of unknowns, and b is the vector of constants.
This document explains the distance formula and how to use it to calculate the distance between points on a Cartesian plane. It provides the steps to use the Pythagorean theorem to derive the distance formula: take the difference between the x-coordinates squared and add it to the difference between the y-coordinates squared, and take the square root of the result. It then works through an example of using the distance formula to calculate the distance between points (3,2) and (8,7), which equals 5 units. The document concludes with practice problems and assignments applying the distance formula.
Group Theory and Its Application: Beamer Presentation (PPT)SIRAJAHMAD36
This document provides an overview of a seminar presentation on group theory and its applications. The presentation covers topics such as the definition of groups, order of groups and group elements, modular arithmetic, subgroups, Lagrange's theorem, and Sylow's theorems. It also discusses some examples of groups and applications of group theory in fields like algebraic topology, number theory, and physics. The presentation aims to introduce fundamental concepts in modern algebra through group theory.
This document contains a semi-detailed lesson plan for teaching dividing rational expressions in Math 8. The lesson plan outlines intended learning outcomes, learning content including the subject matter and references, required materials, and learning experiences such as sample activities, analysis questions, and applications. It provides examples of dividing rational expressions and asks students to divide additional rational expressions as an evaluation of their understanding.
Factorials are the product of all positive integers from a given number down to 1. This document provides examples of calculating factorials such as 4! = 24 and discusses the rules for operations involving factorials such as cancellation rules that apply when dividing factorials. Practice problems are included with explanations to reinforce understanding of how to calculate and operate with factorial expressions.
This document discusses different methods for solving linear equation systems, including LU factorization methods like Doolittle, Crout, and Cholesky decomposition. It provides examples of applying each method to solve systems of equations step-by-step. The Doolittle method assumes the diagonal of the lower triangular matrix L has ones. The Crout method assumes the diagonal of the upper triangular matrix U has ones. The Cholesky method decomposes the matrix into the product of a lower triangular matrix and its transpose. It explains that LU decomposition costs 2n^3/3 flops to decompose the matrix and solve the system, making it faster than directly inverting the matrix.
This document discusses group theory concepts including:
1) Definitions of groups, abelian groups, order of groups and elements.
2) Properties of cyclic groups, including examples like Zn and Z.
3) Introduction to normal subgroups and their properties. Factor groups and homomorphisms are also discussed.
The document defines relations and functions. A relation is a set of ordered pairs, while a function is a special type of relation where each x-value is mapped to only one y-value. The domain is the set of x-values and the range is the set of y-values. Functions can be identified using the vertical line test or by mapping the relation to check if any x-values are mapped to multiple y-values. Evaluating functions involves substituting domain values into the function rule to find the corresponding range values.
This document discusses two types of combinations: combinations with repetition and combinations without repetition. Combinations with repetition allow elements to be chosen multiple times from a set, while combinations without repetition do not allow repeats. It provides examples to illustrate each type, such as how many 2-letter words can be created from 4 letters with repetition (10 combinations) versus without repetition (6 combinations). The key difference between the two types is whether repetition of elements is allowed or not when selecting elements from a set.
Permutations involve arranging objects in a particular order, where order matters. There are a few different formulas to calculate the number of permutations depending on the situation. The basic formula for permutations of n objects taken r at a time is nPr = n!/(n-r)!. Some examples provided calculate the number of permutations when arranging letters, skiers in a competition, and people entering a cave.
The document discusses the standard form of a circle equation and provides examples of writing the equation of various circles given their center and radius. It begins by defining the components of a circle - the center (h,k) and radius r. Then it shows the standard form of a circle equation: (x-h)2 + (y-k)2 = r2. The rest of the document provides the center and radius of several circles and has the reader write out the corresponding standard equation for each one.
The document defines what a group is in mathematics. A group is a set with an operation that is associative, has a neutral element, and where each element has an inverse. Some examples of groups are the integers under addition, rational numbers under addition, and non-zero real numbers under multiplication. Finite groups with a set number of elements, like integers modulo n, are especially important for scientific applications. Not all groups are commutative, as shown by the group of matrices under multiplication.
This document provides an overview of radicals and their properties. It discusses:
- Simplifying radicals using product, quotient, and nth root rules
- Adding and subtracting radicals by combining like terms
- Multiplying and dividing radicals using distribution and rationalizing denominators
- Solving equations containing radicals by isolating radicals and raising both sides to matching powers
- Applications including using the Pythagorean theorem and distance formula to solve problems involving radicals
The document discusses various rules and concepts related to exponents and radicals. It presents examples showing how to simplify expressions using rules such as adding exponents with the same base, distributing exponents, setting exponents of zero equal to one, subtracting exponents with the same base, changing negative exponents to positive forms, and properties of radicals like adding only if they have the same radicand. It emphasizes working through examples as the most important way to understand and apply the rules.
BCA_Semester-I_Mathematics-I_Set theory and functionRai University
This document provides definitions and explanations of key concepts in set theory:
- A set is a collection of well-defined objects or elements. Sets can be finite or infinite.
- Notation involves listing elements within curly brackets. The empty set contains no elements.
- A subset contains elements that are also in another set. The empty set is a subset of all sets.
- Two sets are equal if they contain the same elements. Order refers to the number of elements in a set.
- Proper subsets contain strictly fewer elements than the parent set they are contained within.
This document discusses abstract algebra and its applications. It begins by defining algebra and abstract algebra, which deals with algebraic structures and operations between elements. It then defines some key concepts in abstract algebra like groups, subgroups, cyclic groups, and cosets. It provides examples of dihedral groups and how they appear in nature and corporate logos. Finally, it outlines other applications of abstract algebra in fields like chemistry, cryptography, and solving Rubik's cubes.
The document discusses sets, relations, and functions. It begins by providing examples of well-defined and not well-defined collections to introduce the concept of a set. A set is defined as a collection of well-defined objects. Standard notations for sets are introduced. Sets can be represented using a roster method by listing elements or a set-builder form using a common property. Sets are classified as finite or infinite based on the number of elements, and other set types like the empty set and singleton set are defined. Equal, equivalent, and disjoint sets are also defined.
This document discusses different types of numbers and number systems. It defines natural numbers, integers, rational numbers, irrational numbers, and real numbers. Rational numbers can be written as ratios of integers and have repeating decimals, while irrational numbers have non-repeating decimals. The real numbers consist of rational and irrational numbers. Sets of numbers are also introduced, including unions, intersections, subsets, and the empty set.
This document describes a summer internship project on plotting Cayley graphs for dihedral and alternating groups. It discusses Cayley graphs and provides examples of Cayley graphs for various groups like the dihedral group D3. It also discusses dihedral groups, alternating groups, and permutation groups. The project details section describes the software and hardware requirements for implementing the project, including using Java and Eclipse. It concludes by summarizing how Cayley graphs for dihedral and alternating groups will be plotted using these tools.
This document discusses the group SU(2)xU(1), which describes the electroweak interaction. It first covers relevant group theory concepts like Lie groups and representations. It then explains that SU(2) corresponds to rotations of spinors in real space, and physically represents weak isospin. Together with U(1), SU(2)xU(1) gives rise to the three weak gauge bosons through its symmetry with weak isospin. Representations of these groups relate their mathematical properties to observable physical phenomena.
This document provides information about sets, relations, and functions in mathematics. It begins by giving examples of sets and non-sets to illustrate what makes a collection a well-defined set. It then defines various set concepts like finite and infinite sets, the empty set, singleton sets, equal sets, subsets, unions and intersections of sets. It introduces the concept of relations and functions, defining a function as a special type of relation. It concludes by stating the objectives of learning about sets, relations and functions.
The document discusses sets, relations, and functions. It begins by providing examples of well-defined and not well-defined collections to introduce the concept of a set. A set is defined as a collection of well-defined objects. Standard notations for sets are introduced, such as using capital letters to represent sets and lowercase letters for elements. Two methods for representing sets are described: roster form and set-builder form. The document then classifies sets as finite or infinite, empty, singleton, equal, equivalent, and disjoint sets. Examples are provided to illustrate each concept.
The document discusses several probability concepts including:
1) The probability of outcomes from coin tosses and dice rolls can be calculated by dividing the number of desired outcomes by the total number of possible outcomes.
2) Probability tree diagrams can help calculate probabilities by visually representing the different outcomes and their likelihoods at each step.
3) There are two types of permutations - with and without repetition - which determine whether the same outcome can occur multiple times.
This document provides an introduction to group theory from a physicist's perspective. It defines what a group is, including properties like closure, associativity, identity, and inverse. Examples of important groups in physics are given, including finite groups like Zn and Sn, and continuous groups like SU(n), SO(n), and the Lorentz group. The document outlines topics like discrete and finite groups, representation of groups, Lie groups and algebras, and applications of specific groups like SU(2) and SU(3) to physics.
The document discusses the history and concepts of radicals. It explains that Pythagoras and his followers believed that natural numbers and proportions between natural numbers governed the universe. However, the Pythagorean theorem disproved this by showing the existence of irrational numbers like the square root of 2. The key points are:
- Pythagoras' philosophical theory was disproved by the existence of irrational numbers like the square root of 2 from the Pythagorean theorem.
- The Pythagorean theorem states that the square of the hypotenuse of a right triangle equals the sum of the squares of the other two sides.
- Radicals can be used to express solutions to equations and powers with fractional
The document discusses fuzzy logic and fuzzy sets. It defines fuzzy sets as sets with non-crisp boundaries where elements have degrees of membership between 0 and 1 rather than simply belonging or not belonging. It outlines some key concepts of fuzzy sets including membership functions, basic types of fuzzy sets over discrete and continuous universes, and set-theoretic operations like union, intersection, and complement for fuzzy sets.
5. Limit Fungsi yang menjadi Aljabar.pptxBanjarMasin4
This document discusses limits of algebraic functions in mathematics education. It begins with definitions of one-sided limits and general limits. It then covers theorems related to limits, including the Limit Theorems, Substitution Theorem, Squeeze Theorem, and the relationship between continuity and limits of functions. Examples are provided to demonstrate applying the theorems to calculate limits. The document provides foundations for understanding limits of functions through rigorous definitions and theorems.
The document discusses backtracking as a general method for solving problems that involve systematically trying possibilities and abandoning partial solutions when they cannot possibly lead to a complete solution. It provides examples of applications of backtracking including the n-queens problem, sum of subsets problem, graph coloring, and finding Hamiltonian cycles in graphs. The general backtracking algorithm and terminology used are described. Specific algorithms for solving the n-queens problem and sum of subsets problem using backtracking are also presented.
- The document discusses perfect squares, square roots, cubes, and nth roots.
- It provides examples of finding square roots and cube roots of various numbers.
- Key formulas introduced include the volume of a sphere and estimating distance to the horizon.
The document discusses multiplying matrices by numbers and scalar multiplication of matrices. It states that to multiply a matrix by a number, each element of the matrix is multiplied by that number. Scalar multiplication involves multiplying a matrix by a real number called a scalar, where each element of the resulting matrix is that scalar multiplied by the corresponding original element. The document also provides examples of adding, subtracting and scalar multiplying matrices.
To multiply a matrix by a number, multiply each element of the matrix by that number. For example, to multiply a 2x2 matrix by 2, multiply each element by 2. Similarly, to multiply by 3, multiply each element by 3. This process works for any real, imaginary, fractional or decimal number. Matrix addition and subtraction can only be done on matrices with the same dimensions, by adding or subtracting the corresponding elements. For scalar multiplication, each element of the product matrix is the scalar multiplied by the corresponding element of the original matrix.
Similar to Introduction to Groups and Permutation Groups (20)
Green chemistry, Its Applications and BenefitsAmit Amola
Green chemistry is the design of chemical products and processes to reduce or eliminate the use and generation of hazardous substances. It was formally established 15 years ago by the EPA in response to pollution regulations. The key principles of green chemistry developed by Anastas and Warner include preventing waste through inherently safer design of synthesis processes and products. Examples show how green chemistry has led to replacement of hazardous chemicals like phosgene and solvents like benzene with safer alternatives like solid-state synthesis routes and ionic liquids.
The document appears to be a reference document owned by Amit Amola containing various symbols and numbers across 24 pages. Each page is marked as the property of Amit Amola and for reference and consent only. The content includes overview, symbols like circles and checks, and various numeric values.
English Translation- Machine Translation Amit Amola
The document discusses machine translation and provides examples of machine translation programs and services. It then provides a sample text in English and the machine translation of the text into Hindi. Several excerpts from the English text are shown side by side with their Hindi translations. The summary analyzes some differences between the original English text and the Hindi translations, noting issues like loss of meaning, changes in grammar and parts of speech, and failures to accurately translate metaphors.
Functions and its Applications in MathematicsAmit Amola
A function is a relation between a set of inputs and set of outputs where each input is related to exactly one output. An example is given of a function that relates shapes to colors, where each shape maps to one unique color. A function can be written as a set of ordered pairs, where the input comes first and the output second. The domain is the set of inputs, the codomain is the set of possible outputs, and the range is the set of outputs the function actually produces. A function is one-to-one if no two distinct inputs map to the same output, and onto if every element in the codomain is mapped to by at least one input.
The document discusses line integrals in the complex plane. It defines line integrals, shows how complex line integrals are equivalent to two real line integrals, and reviews how to parameterize curves to evaluate line integrals. It also covers Cauchy's theorem, which states that the line integral of an analytic function around a closed curve in its domain is zero. The fundamental theorem of calculus for complex variables and the Cauchy integral formula are also summarized.
Amit Amola successfully completed a winter 2014 internship program with Eagles' Cocoon from December 8, 2014 to January 20, 2015. During his internship, he helped carry out various digital marketing initiatives to expand the global reach of India For Transformation, one of Eagles' Cocoon's portfolio organizations. He demonstrated dedication to his work and an ability to navigate ambiguous situations with confidence. Eagles' Cocoon wishes him luck in future endeavors.
Amit Amola successfully completed a six-month internship with CampusDope from October 2014 to March 2015. During this time, he demonstrated strong writing and marketing skills, perseverance, and passion for his work. CampusDope appreciated his intellectually strong acumen and ability to deliver quality work on time. They enjoyed working with him and hope to stay in touch in the future.
Amit Amola has received an internship completion certificate from CampusDope for successfully completing a six-month internship starting in October 2014. During his internship, Amit demonstrated strong content writing and marketing skills and a passion for his work, which the co-founders of CampusDope admired. CampusDope congratulates Amit on the successful completion of his internship.
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
How to Manage Your Lost Opportunities in Odoo 17 CRMCeline George
Odoo 17 CRM allows us to track why we lose sales opportunities with "Lost Reasons." This helps analyze our sales process and identify areas for improvement. Here's how to configure lost reasons in Odoo 17 CRM
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বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
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Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
Your Skill Boost Masterclass: Strategies for Effective Upskilling
Introduction to Groups and Permutation Groups
1. Property of Amit Amola. Should be used only for reference and with consent.
We all know in full depth what groups are and what are their properties.
A group is a nonempty set together with a binary operation in such a way
that there is an identity element in that group, there is associativity in its
elements, every element has an inverse, and any pair of element can be
obtained without going outside the set (closure). With this definition now
let us try to understand Groups in more detailed manner with a basic
example.
Take this hexagonal “Ok” sign for an example:-
Let’s see that in how many ways we can reposition this hexagonal “OK”
sign-
By 60° rotation either towards left or right we can find six other positions.
The next 60° rotation would give back to us the initial position itself. Now
we shall check the other positions which we can get by flipping or
reflecting the hexagon through its side’s midpoints and through its vertices.
Abstract Algebra
Assignment
R0 R60 R120 R180 R240 R300
V3V2V1 M1 M2 M3
2. Property of Amit Amola. Should be used only for reference and with consent.
These are the only 12 positions which we can find or form by repositioning
or reflecting the hexagon. Now we can see that every motion which we
perform will give any of these 12 outcomes only. For example let’s rotate
the initial position by 120° and then reflect it through first of the three
main vertices.
We can clearly see that the final image is V3 only. Which means V1R120 = V2
So these 12 motions along with their compositions form a mathematical
system called the Dihedral Group. In this case as the number of elements or
total number of motions are 12 so the order of this Dihedral Group is 12 i.e.
this is the Dihedral Group of order 12. And it is denoted as D6. Now we shall
understand what groups exactly are with the help of the example we have
taken above.
For this we use a much simpler approach by constructing an operation
table or which we refer to as Cayley Table to see and observe the different
composition of motions which we can find.
R0 R60 R120 R180 R240 R300 V1 V2 V3 M1 M2 M3
R0 R0 R60 R120 R180 R240 R300 V1 V2 V3 M1 M2 M3
R60 R60 R120 R180 R240 R300 R0 M1 M2 M3 V2 V3 V1
R120 R120 R180 R240 R300 R0 R60 V2 V3 V1 M2 M3 M1
R180 R180 R240 R300 R0 R60 R120 M2 M3 M1 V3 V1 V2
R240 R240 R300 R0 R60 R120 R180 V3 V1 V2 M3 M1 M2
R300 R300 R0 R60 R120 R180 R240 M3 M1 M2 V1 V2 V3
V1 V1 M1 V2 M2 V3 M3 R120 R240 R0 R180 R300 R60
V2 V2 M2 V3 M3 V1 M1 R240 R0 R120 R300 R60 R180
V3 V3 M3 V1 M1 V2 M2 R0 R120 R240 R60 R180 R300
M1 M1 V2 M2 V3 M3 V1 R180 R300 R60 R240 R0 R120
M2 M2 V3 M3 V1 M1 V2 R300 R60 R180 R0 R120 R240
M3 M3 V1 M1 V2 M2 V3 R60 R180 R300 R120 R240 R0
R120 V1
3. Property of Amit Amola. Should be used only for reference and with consent.
These are the inferences we can take out from this Cayley table:-
We can see in this table that all the outcomes are no different than the
already known motions. This means that any motion which we carry out
will result in these 12 outcomes only; means if P and Q are in D6 then PQ
is also in D6. This property is referred to as closure, and this is one of the
necessary requirement for a particular set to be defined as a group.
We also see that in this set of motions i.e. D6 there exist an identity
element (here R0) such that if there is any other element let’s say P of D6,
then R0P = PR0 =P. So there must be an identity element in a set for it
being called as a group.
Now we also see that for each element P in D6 there is exactly one
element Q in D6 such that PQ = QP = R0. Here P and Q are known as
inverse of each other. For example here in our table inverse of R240 is
R120 and vice versa and like R180 and V2 are their own inverses. Again this
is an important condition for a set to be referred as group.
One more condition which is compulsory for a set to be referred as
group is associativity in its elements; i.e. (pq)r = p(qr) where p,q and r є
D6. There are 123= 1728 possible choices of p,q and r in D6 in which we
have to check associativity. Here it is not humanly possible but as we
know that the given motions are functions itself and the operation here
is of function composition. And since function composition is always
associative, so we take it as obvious that all the elements of our dihedral
group undergo associativity.
One another keen observation here shows that every element of D6
appears exactly once in each row and column. This feature is something
that all groups must possess and moreover we should keep this fact in
mind while making Cayley table.
And the last observation is about that if the group is commutative or not;
i.e. if it’s any choice of group of elements let’s say P and Q are such that
PQ = QP then the group is commutative which is referred to as Abelian
groups. And if they aren’t commutative then they are just called Non-
abelian groups.
4. Property of Amit Amola. Should be used only for reference and with consent.
So we just saw the three necessary conditions which a non-empty set must
possess to be referred as a group beside it being closure; which namely are
associativity in its elements, presence of an identity element in that set &
availability of inverses of every element of the set.
Now let’s discuss Dihedral Groups in detail.
Dihedral Groups
In simple way a Dihedral Group is a group whose elements are symmetries
of a regular polygon and these symmetries are obtained by rotations and
reflections of the polygon from its vertices and mid-points of its sides.
Dihedral Groups are considered to be the simplest example of finite groups.
We can construct the similar symmetrical figures for other polygons too
with sides ≥3. And these groups are denoted as Dn and is called dihedral
group of order 2n, where 2n are the total number of elements in that group.
If the polygon has n sides, then it has 2n symmetries: n rotational
symmetries and n reflective symmetries. Let’s take another example which
I’ve taken from Wikipedia. Here is an Octagonal “Stop” sign which is
dihedral group of order (2x8) =16. So here are the 16 symmetrical figures
of Octagonal “Stop” sign.
The first row shows the effect of the
eight rotations, and the second row
shows the effect of the eight reflections.
Similarly a snowflake has D6 dihedral
symmetry, the same as a regular
hexagon.
5. Property of Amit Amola. Should be used only for reference and with consent.
Dihedral Groups are everywhere around us. Take example of various
company’s logos.
Like logo of Chrysler’s logo has D5 symmetry.
Mercedes-Benz logo has D3 as symmetry group.
6. Property of Amit Amola. Should be used only for reference and with consent.
Either take human made Rose Windows in a church as an example:-
Like this Rose Window shows Group of Symmetry of order 18, i.e. D9.
Or take for example Crop Circles which are presumed to be made by alien.
7. Property of Amit Amola. Should be used only for reference and with consent.
Well that was all about Dihedral Groups. Now before we talk about
Permutation Groups, let’s give a quick look on some other important terms
in a simplified manner.
Order of a Group- The number of elements in a group are called that group’s
order. And is denoted as |G| if we have to denote the order of the group G.
For example |D6|= 12.
Order of an element- The order of an element say m in a group say M is the
smallest positive integer n s.t. mn = e. If no such n exists then we say m has
infinite order. The order of m is denoted as |m|. For example order of R60 in
D6 is 6; i.e. | R60|= 6.
Subgroups- If there’s a subset H of a group G which is itself a group under
the operation of G, we say H is a subgroup of G. We denote it as H<G. There
are various methods to test whether the subset of a group taken under
consideration is a subgroup or not. Here I am just mentioning what those
tests are:-
One-Step Subgroup Test
Two-Step Subgroup Test
Finite Subgroup Test
Cyclic Groups- A group say F is called cyclic if there is an element ‘a’ in F s.t.
F= {an | n є Z}. And such an element ‘a’ is called a generator of F. We also
write F as F= ⟨a⟩.
Now knowing these important terms we shall proceed to Permutation
Groups.
Permutation Groups
These are one of the important class of groups. One reason for their
importance is that most of the groups can be represented as a group of
permutations on a suitable set. (Exception include Quaternion Groups)
Now let’s get to know what Permutation Groups are-
A permutation group is a finite group let’s say G whose elements are
permutations of a given set and whose group operation is composition of
permutations in G. There is one-to-one mapping from G onto G. Well to
explain it better let’s see a very interesting example.
8. Property of Amit Amola. Should be used only for reference and with consent.
Let’s consider a set X containing 3 objects, say a triangle, a circle and a
square. A permutation of X = {△, O, □} might send for example
△ △, O □, □ O
and we observe that what just did is exactly to define a bijection on the set
X, namely a map σ: X X defined as
σ (△) = △, σ ( O) = □, σ (□) = O
Picture representation:-
X= { , , }
Now defining an arbitrary bijection:-
Now such function can be denoted through this notation
which is known as two-row form:-
( )
And if we denote these symbols as numbers then we can
also write this as:
(
𝟏 𝟐 𝟑
𝟏 𝟑 𝟐
)
9. Property of Amit Amola. Should be used only for reference and with consent.
So just above we saw one form of set X in which we can write its elements.
There are 5 other ways. For example let’s take set Y= {1, 2, 3} and let’s
discuss its 6 six permutation forms. I am using two row form to represent
these forms:
a= (
1 2 3
1 2 3
) b= (
1 2 3
2 3 1
) c= (
1 2 3
3 1 2
)
d= (
1 2 3
2 1 3
) e= (
1 2 3
3 2 1
) f= (
1 2 3
1 3 2
)
So these are the six permutation forms which we can make out from set Y.
And these are the six elements of its permutation group. Such kind of
groups are also known as Symmetry Groups and denoted as Sn.
Here we can also see that (b.f)≠(f.b), so that S3 is not an abelian group.
Now before I explain about this group’s identity element and the inverse, I
ought to introduce about composition of the elements of a permutation
group. In composing permutations we always follow the same convention
we use in composing any other mappings: read from right to left. For
example let,
σ= (
1 2 3
3 1 5
4 5
4 2
) λ= (
1 2 3
4 3 1
4 5
2 5
)
σλ = (
1 2 3
3 1 5
4 5
4 2
) (
1 2 3
4 3 1
4 5
2 5
) = (
1 2 3
4 5 3
4 5
1 2
)
We see that finally on right we have 4 below 1, it’s so because
(σλ)(1)= σ(λ)(1) = σ(4)= 4, so σλ sends 1 to 4. And other elements are
obtained in similar manner.
Now as we know how composition in elements of permutation group
works, so let’s talk about identity element and inverse of S3 group. An
identity element should be of a kind such that if any element of S3 say ‘m’ is
operated with it then it gives ‘m’ only as a result.
So here a= (
1 2 3
1 2 3
) is the identity element.
Let’s say we take element ‘b’, so we can see that a.b=b.a=a.
10. Property of Amit Amola. Should be used only for reference and with consent.
Now let’s talk about inverse of these elements. Let’s find inverse of element
c.
c= (
1 2 3
3 1 2
) c-1= (
1 2 3
2 3 1
)
Thus, the inverse of an element is obtained by reading from the bottom
entry to the top entry rather than from top to bottom: if 1 appears beneath
3 in c then 3 appears beneath 1 in c-1.
So this was about symmetric group S3. Let’s talk little more about
Symmetric groups in general.
Symmetric Groups (Sn)
Let A= {1, 2, 3, …, n}. The set of all permutations of A is called the
symmetric group of degree n and is denoted by Sn. Elements of Sn have the
form
є= [
1 2
є(1) є(2)
… … … …
𝑛
є(n)]
It is easy to compute the order of Sn. There are n choices of є(1). Once є(1)
has been determined, there are (n-1) possibilities for є(2)[since є is one-to-
one, we must have є(1)≠ є(2)]. After choosing є(2), there are exactly (n-2)
possibilities for є(3). Continuing along in this fashion, we see that Sn must
have n(n-1)…3.2.1= n! elements.
Now, since S1 = {(1)} then S1 with respect to composition is commutative.
Similarly, since
[
1 2
1 2
] [
1 2
2 1
]= [
1 2
2 1
] [
1 2
1 2
]
then S2 = {[
1 2
1 2
] [
1 2
2 1
]}is also Abelian.
Unfortunately, this is not true anymore for |S| > 2.
We will now prove that Sn is not abelian when n≥3.
11. Property of Amit Amola. Should be used only for reference and with consent.
Theorem- Sn is non-Abelian for n≥3.
Proof: All that we need to do here is to find two permutations σ and λ in Sn
with n≥3 such that (σ ○ λ) ≠ (λ ○ σ). Indeed, consider the permutations
σ= (
1 2
1 3
3 4 5
2 4 5
… … …
𝑛
𝑛
) and λ= (
1 2
3 2
3 4 5
1 4 5
… … …
𝑛
𝑛
)
we know that of course such two elements will always exist in a symmetric
group where n≥3. And moreover we can see that
σ ○ λ= (
1 2
2 3
3 4 5
1 4 5
… … …
𝑛
𝑛
) and λ ○ σ= (
1 2
3 1
3 4 5
2 4 5
… … …
𝑛
𝑛
)
i.e. σ ○ λ≠ λ ○ σ
Hence for n≥3, Sn is always non-abelian.
Cycle Notation for Permutations
There’s one more way we write or denote the elements of permutation. It is
called as Cycle Notation. The cycle notation for permutations can be
thought as a condensed way to write permutations. Here is how it works.
Let α є Sn be the permutation
and α(a1) = a2, α(a2) = a3, · · · , α(ak) = a1
i.e. α follows the circle pattern
Such a permutation is called a cycle of length k or simply a k-cycle. We will
write
α= (a1a2a3……ak)
The cycle notation is read from left to right, it says α takes a1 into a2, a2 into
a3, etc., and finally ak, the last symbol, into a1, the first symbol.
a1
ak
a2
a3
..
.
..
...
..
.
12. Property of Amit Amola. Should be used only for reference and with consent.
Let’s see how this cycle notation works with an example:
Let say we have an element as
α= (
1 2
1 6
3 4 5
3 7 5
6
4
7
2
)
Some points to get acquainted with:
So this element α can also be written in cycle notation as
α= (1)(2647)(3)(5) or just by a 4-cycle notation (2647)
It is customary to omit the terms whose image is same as their own.
Moreover this notation can be used in many ways.
α= (2647)
= (6472)
= (4726)
= (7264)
This thing above shows us that a k-cycle can be written in k different
ways, since
(a1a2a3……ak)= (a2a3 a4……ak a1) = … = (aka1 a2……ak-1)
We can easily find the inverse of a cycle. Since α(ak)= ak+1 implies
α-1(ak+1)= ak , we only need to reverse the order of the cyclic pattern. For
example,
(2647)-1 = (7462)
Now let’s get acquainted with how the multiplication of cycles are done.
Multiplication of cycles- Multiplication of cycles is performed by applying
the right permutation first.
Consider the product in α= (12)(3)(45) and β= (153)(24) and α β=
(12)(3)(45) (153)(24)
13. Property of Amit Amola. Should be used only for reference and with consent.
Now from reading right to left,
1 1 5 4 4 4
So 1 4.
Now
4 2 2 2 2 1
So 4 1.
Now
2 4 4 5 5 5
So 2 5.
Now
5 5 3 3 3 3
So 5 3
Now
3 3 1 1 1 2
So 3 2. So we can see that this multiplication can be represented as
αβ= (14)(253)
i.e. αβ= (12)(3)(45) (153)(24)= (14)(253)
We just saw that in the original multiplication term there were some cycles
with same element. But in the final result there is no such thing. So those
two cycles are called disjoint cycles.
There are two theorems which are associated with disjoint cycles. Let’s see
what they say.
Theorem: If α and β are disjoint cycles then αβ = βα.
Proof-
Indeed, since the cycles α and β are disjoint, each element moved by α is
fixed by β and vice versa. Let α = (a1a2 · · · as) and β = (b1b2 · · · bt) where
{a1, a2, · · · , as} ∩ {b1, b2, · · · , bt} = ∅.
(i) Let 1 ≤ k ≤ s. Then
(αβ)(ak) = α(β(ak)) = α(ak) = ak+1
and
(βα)(ak) = β(α(ak)) = β(ak) = ak+1
(ii) Let 1 ≤ k ≤ t. Then
(αβ)(bk) = α(β(bk)) = α(bk+1) = bk+1
and
(βα)(bk) = β(α(bk)) = β(bk) = bk +1
14. Property of Amit Amola. Should be used only for reference and with consent.
(iii) Let 1 ≤ m ≤ n and m ∉{a1, a, … as, b1, b2, … , bt}. Then
(αβ)(m) = α(β(m)) = α(m) = m
and
(βα)(m) = β(α(m)) = β(m) = m
It follows from (i), (ii), and (iii) that αβ = βα.
So that was theorem which says that if α and β are disjoint cycles then αβ =
βα. Now we have one more theorem which tells that every permutation of a
finite set can be written as a cycle or product of disjoint cycle.
Theorem: Every permutation of a finite set can be written as a
cycle or as a product of disjoint cycles.
Proof-
Let α be a permutation on A = {1, 2, … ,n}. To write α in disjoint cycle form,
we start by choosing any member of A, say a1 and let
a2= α(a1), a3= α(α(a1)) = α2(a1),
and so on, until we arrive at a1 = αm(a1) for some m. We know such an m
exists because the sequence a1, α(a1), α2(a1), . . . must be finite; so there
must eventually be a repetition, say αi(a1) = αj(a1) for some i and j with
i < j. Then a1= αm(a1), where m = (j – i). We express this relationship
among a1,a2, . . . . . ., am as
α= (a1,a2, . . . . . ., am) . . .
The ellipsis at the end indicate the possibility that we may not have
exhausted the set A in this process. In such a case, we merely choose an
element b1 of A not appearing in the first cycle and proceed to create a new
cycle as before. That is, we let b2 = α(b1), b3= α2(b1), and so on, until we
reach b1 = αk(b1) for some k. This new cycle will have no elements in
common with the previously constructed cycle. For, if so, then
αi(a1) = αj(b1) for some i and j. But then αi-j(a1)= b1 and therefore b1= at for
some t. This contradicts the way b1 was chosen. Continuing this process
until we run out of elements of A, our permutation will appear as
α= (a1,a2, . . . , am)( b1,b2, . . . , bk) . . . (c1,c2, . . . , cs).
In this way, we see that every permutation can be written as a product of
disjoint cycles.
15. Property of Amit Amola. Should be used only for reference and with consent.
Now there’s one more theorem which tells us about order of a permutation.
Theorem: The order of a permutation of a finite set written in
disjoint cycle form is the LCM of the lengths of the cycles.
Proof-
We can easily see that a cycle of length n has order n.
Now let’s suppose that α and β are two disjoint cycles of lengths m and n
respectively, and let k be the least common multiple of m and n. Now we
have read a theorem earlier (not proved in this article) which is stated as-
Let G be a group and let ‘a’ be an element of order n in G. If ak = e,
then n divides k.
So from this theorem it follows that both αk and βk are the identity
permutation ε and, since α and β commute, (αβ)k = αkβk is also the identity.
Thus, we know by the corollary to the above theorem (ak = e implies that
|a| divides k) that the order of αβ(let’s say it t) must divide k. But then
(αβ)t= αtβt = ε, so that αt = β-t. However, it is clear that if α and β have no
common symbol, the same is true for αt and β-t, since raising a cycle to a
power does not introduce new symbols. But if αt and β-t are equal and have
no common symbols, they must both be the identity, because every symbol
in αt is fixed by β-t and vice-versa (a symbol not appearing in a permutation
is fixed by the permutation). It follows, then, that both m and n must divide
t. This means that k, the least common multiple of m and n, divides t also.
This shows k = t.
Thus far, we have proved that the theorem is true in the cases where
the permutation is a single cycle or a product of two disjoint cycles. The
general case involving more than two cycles can be handled in similar
fashion.
16. Property of Amit Amola. Should be used only for reference and with consent.
Now there’s an another theorem which is like this-
Theorem: Every permutation in Sn, n > 1, is a product of 2-cycles.
Proof:
Note that the identity can be written as (12)(12).
A k-cycle (a1a2 . . . ak) can be written as
(a1a2 . . . ak) = (a1ak)(a1ak−1) · · · (a1a3)(a1a2).
Since any permutation can be written as a product of disjoint cycles, we can
decompose any permutation into a product of transpositions by
decomposing each disjoint cycle in the product.
Examples:
(12345) = (15)(14)(13)(12) or (54) (52) (21) (25) (23) (13).
(1632)(457) = (47) (45) (12) (13) (16).
One more interesting theorem tell us that:
Theorem: If a permutation α can be expressed as a product of even
number of 2-cycles, then every decomposition of α into a product
of 2-cycles must have an even number of 2-cycles. In symbols, if
α = β1β2 . . . βr and α = γ1γ2 . . . γs,
where the β’s and the γ’s are 2-cycles, then r and s are both even or
odd.
Proof-
Observe that β1β2 . . . βr = γ1γ2 . . . γs implies
ε = γ1γ2 . . . γs βr
-1 . . . β2-1β1-1
= γ1γ2 . . . γs βr . . . β2β1,
since a 2-cycle is its own inverse. Thus, the theorem above guarantees that
s+r is even. It follows that r and s are both even or both odd.
And this leads to these definitions;
A permutation that can be expressed as a product of an even number of 2-
cycles is called an even permutation. A permutation that can be expressed
as a product of an odd number of 2-cycles is called an odd permutation.
17. Property of Amit Amola. Should be used only for reference and with consent.
These above two theorems show that every permutation can be without
any doubt classified as either even or odd, but not both.
This observation has a very big significance. In next theorem i’ll show that
why we needed the above observations.
Theorem: Even permutations form a group
or
The set of even permutations in Sn forms a subgroup of Sn.
Proof-
These set of even permutations occur so often that we refer to them by a
special name as Alternating Group of Degree n or An.
Now applying the finite subgroup test to An:
We apply the Finite Subgroup Test to An:
As per finite subgroup test we need to prove that a-1∈ An whenever a∈An.
We can see e = (12)(12) is an even permutation of Sn. So e ∈ An.
And we know any 2-cycle permutation element γ ∈ An will have γ-1 in An
only.
Now we also see that whether or not it satisfies closure property. Let
α,β ∈ An. Then α is a product of an even number, say 2k, of 2-cycles and β is
a product of an even number, say 2l, of 2-cycles.
Then αβ is a product of an even number, 2(k+l), of 2-cycles i.e. even
permutation. Hence αβ∈ An.
Hence this proves that the set of even permutations in Sn forms a subgroup
of Sn.
Now the next result shows that exactly half of the elements of Sn(n>1) are
even permutations.
18. Property of Amit Amola. Should be used only for reference and with consent.
Theorem: For n>1, An has order n!/2.
Proof-
We know that cycle (12) denotes identity. Now for each odd permutation σ,
the permutation (12)σ is even. Thus there are at least as many even
permutations as there are odd ones. On the other hand, for each even
permutation ϕ the permutation (12) ϕ is odd. So, there are at least as many
odd permutations as there are even ones. It
follows that there are an equal number of even
and odd permutations.
Since |Sn|= n!., thus we have |An|=n!/2.
Example of use of Alternating Groups:
The 12 rotations of a Tetrahedron can be
conveniently described with the elements of A4.
19. Property of Amit Amola. Should be used only for reference and with consent.
Finally a quick look at the history and development of permutation groups.
History of Permutation Groups
The study of groups originally grew out of an
understanding of permutation
groups. Permutations had themselves been
intensively studied by Lagrange in 1770 in his
work on the algebraic solutions of polynomial
equations.
This subject flourished and by the mid-19th
century a well-developed theory of permutation
groups existed, codified by Camille Jordan in his
book of 1870. Jordan's book was, in turn, based on
the papers that were left by Évariste Galois in 1832.
When Cayley introduced the concept of an
abstract group, it was not immediately clear
whether or not this was a larger collection of
objects than the known permutation groups
(which had a definition different from the
modern one). Cayley went on to prove that the
two concepts were equivalent in Cayley's
theorem.
Joseph Louis Lagrange
Camille Jordan
Évariste Galois
Arthur Cayley
20. Property of Amit Amola. Should be used only for reference and with consent.
Another classical text containing several chapters on permutation groups
is Burnside's Theory of Groups of Finite Order of 1911. The first half of the
twentieth century was a fallow period in the study of group theory in
general, but interest in permutation groups was revived in the 1950s by
H. Wielandt whose German lecture notes were reprinted as Finite
Permutation Groups in 1964.
So that’s all I covered in this article. We talked about Groups by giving
example of Dihedral Groups and then discussed them in detail. Then in
between we also talked about some important terms and then we moved
ahead to Permutation Groups where we first introduced about Permutation
Groups and then also introduced Symmetry Groups. We also saw many
theorems and observations and made inferences out of them. So that’s
what we learnt about in this article.
The End
William_Burnside
Helmut Wielandt
21. Property of Amit Amola. Should be used only for reference and with consent.
Reference:
Books:
Contemporary Abstract Algebra (4th ed.) by Joseph A. Gallian
The Mathematics of Identification Numbers by Joseph A. Gallian
A First Course in Abstract Algebra with Applications (3rd ed.) by
Rotman, Joseph J
Abstract Algebra (3rd ed.) by Dummit, David S., Foote, Richard M.
Symmetry Groups and Their Applications by Miller, Willard Jr.
Symmetry by H. Weyl
Websites:
Mathigon- http://world.mathigon.org/
Wikipedia- http://en.wikipedia.org/
Priceton University - www.princeton.edu/
http://math.berkeley.edu/
http://mathworld.wolfram.com/
www.google.com