This document discusses the probability that an element of a metacyclic 2-group of positive type fixes a set. It provides background on metacyclic groups and previous work calculating fixing probabilities for other groups. The main results of the paper calculate the fixing probability for positive type metacyclic 2-groups and apply the results to generalized conjugacy class graphs.
1. The document discusses the probability that an element of a metacyclic 2-group of positive type fixes a set. It defines metacyclic groups and the concept of a group acting on a set.
2. Previous work on the commutativity degree and the probability that a group element fixes a set is summarized. It was shown that this probability equals the number of conjugacy classes divided by the total number of sets.
3. The main results compute the probability that an element of a metacyclic 2-group of positive type fixes a set, by considering the action of the group on the set of commuting element pairs through conjugation.
sarminIJMA1-4-2015 forth paper after been publishingMustafa El-sanfaz
This document discusses the probability that a group element fixes a set and its application to generalized conjugacy class graphs. It begins with background on commutativity degree and graph theory concepts. It then reviews previous work calculating the probability for various groups and defining generalized conjugacy class graphs. The main results calculate the probability for semi-dihedral and quasi-dihedral groups as 1/2 and 1/3 respectively, and determine that the corresponding generalized conjugacy class graphs are K2 and Ke (empty graph).
This document discusses the probability that an element of a metacyclic 2-group fixes a set. It begins by providing background information on metacyclic groups and related concepts such as commutativity degree. It then summarizes previous works that have studied the probability of a group element fixing a set. The main results presented calculate the probability for various metacyclic 2-groups of negative type with nilpotency class of at least two. In particular, it determines the probability for metacyclic 2-groups of negative type with nilpotency class of at least three. It also calculates the probability for other specific metacyclic 2-group structures.
1) The document discusses the probability that an element of a dihedral group fixes a set under group actions like conjugation.
2) It defines the probability as the number of orbits of the set divided by the size of the set.
3) For a dihedral group G of order 2n, the probability is 4/3n^2 if n and n/2 are even, and 3/3n^2 if n is even and n/2 is odd.
This document discusses the probability that an element of a metacyclic 3-group of negative type fixes a set and its orbit graph. It begins by providing background on commutativity degree, metacyclic p-groups, and basic graph theory concepts. Previous related work calculating the probability that an element fixes a set is summarized. The document then presents the main results, which are computing the probability that an element of a metacyclic 3-group of negative type fixes a set, and applying this to construct the orbit graph.
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.
1) The document discusses testing modifications to general relativity (GR) to explain the observed accelerated expansion of the universe. Two representative modified gravity (MG) models are studied: f(R) theories and the Bean-Tangmatitham parametrization.
2) These MG models modify the gravitational potentials in GR through additional parameters that can generate different gravitational behaviors, including those described by GR.
3) The MG parameters are constrained using CosmoSIS, a cosmological parameter estimation code, with data from the Cosmic Microwave Background and weak lensing surveys. Estimates of the MG parameters suggest no deviation from GR predictions.
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.
1. The document discusses the probability that an element of a metacyclic 2-group of positive type fixes a set. It defines metacyclic groups and the concept of a group acting on a set.
2. Previous work on the commutativity degree and the probability that a group element fixes a set is summarized. It was shown that this probability equals the number of conjugacy classes divided by the total number of sets.
3. The main results compute the probability that an element of a metacyclic 2-group of positive type fixes a set, by considering the action of the group on the set of commuting element pairs through conjugation.
sarminIJMA1-4-2015 forth paper after been publishingMustafa El-sanfaz
This document discusses the probability that a group element fixes a set and its application to generalized conjugacy class graphs. It begins with background on commutativity degree and graph theory concepts. It then reviews previous work calculating the probability for various groups and defining generalized conjugacy class graphs. The main results calculate the probability for semi-dihedral and quasi-dihedral groups as 1/2 and 1/3 respectively, and determine that the corresponding generalized conjugacy class graphs are K2 and Ke (empty graph).
This document discusses the probability that an element of a metacyclic 2-group fixes a set. It begins by providing background information on metacyclic groups and related concepts such as commutativity degree. It then summarizes previous works that have studied the probability of a group element fixing a set. The main results presented calculate the probability for various metacyclic 2-groups of negative type with nilpotency class of at least two. In particular, it determines the probability for metacyclic 2-groups of negative type with nilpotency class of at least three. It also calculates the probability for other specific metacyclic 2-group structures.
1) The document discusses the probability that an element of a dihedral group fixes a set under group actions like conjugation.
2) It defines the probability as the number of orbits of the set divided by the size of the set.
3) For a dihedral group G of order 2n, the probability is 4/3n^2 if n and n/2 are even, and 3/3n^2 if n is even and n/2 is odd.
This document discusses the probability that an element of a metacyclic 3-group of negative type fixes a set and its orbit graph. It begins by providing background on commutativity degree, metacyclic p-groups, and basic graph theory concepts. Previous related work calculating the probability that an element fixes a set is summarized. The document then presents the main results, which are computing the probability that an element of a metacyclic 3-group of negative type fixes a set, and applying this to construct the orbit graph.
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.
1) The document discusses testing modifications to general relativity (GR) to explain the observed accelerated expansion of the universe. Two representative modified gravity (MG) models are studied: f(R) theories and the Bean-Tangmatitham parametrization.
2) These MG models modify the gravitational potentials in GR through additional parameters that can generate different gravitational behaviors, including those described by GR.
3) The MG parameters are constrained using CosmoSIS, a cosmological parameter estimation code, with data from the Cosmic Microwave Background and weak lensing surveys. Estimates of the MG parameters suggest no deviation from GR predictions.
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.
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.
The document provides notes on group theory. It discusses the definition of groups and examples of groups such as (Z, +), (Q, ×), and Sn. Properties of groups like Lagrange's theorem and criteria for subgroups are also covered. The notes then discuss symmetry groups, defining isometries of R2 and showing that the set of isometries forms a group. Symmetry groups G(Π) of objects Π in R2 are introduced and shown to be subgroups. Specific examples of symmetry groups like those of triangles, squares, regular n-gons, and infinite strips are analyzed. Finally, the concept of group isomorphism is defined and examples are given to illustrate isomorphic groups.
IRJET - Triple Factorization of Non-Abelian Groups by Two Minimal SubgroupsIRJET Journal
This document presents research on the triple factorization of non-abelian groups by two minimal subgroups. It begins with an abstract discussing the triple factorization of a group G as G = ABA, where A and B are proper subgroups of G. It then studies the triple factorizations of two classes of non-abelian finite groups: the dihedral groups D2n and the projective special linear groups PSL(2,2n). Several lemmas and a main theorem are presented regarding the triple factorizations and related rank-two coset geometries of these groups. In particular, the theorem discusses conditions for the existence of non-degenerate triple factorizations of D2n and properties of the associated rank-two coset
On The Properties of Finite Nonabelian Groups with Perfect Square Roots Using...inventionjournals
In this paper we determine some internal properties of non abelian groups where the centre Z(G) takes its maximum size. With this restriction we discover that the group G is necessarily a group with perfect square root as shown in our results. We also show that the property of Z(G) is inherited by direct product of groups. Furthermore groups whose centre is as large as possible were constructed.
BCA_Semester-II-Discrete Mathematics_unit-i Group theoryRai University
This document provides an introduction to group theory, including definitions of key concepts such as binary operations, groups, abelian groups, subgroups, cyclic groups, and permutation groups. It defines what constitutes a group and subgroup. Theorems covered include Lagrange's theorem about the order of subgroups dividing the group order, and that every subgroup of a cyclic group is cyclic. Examples are provided of groups defined by binary operations and permutation groups. Exercises at the end involve applying the concepts to specific groups and proving properties of groups.
This document provides definitions and propositions related to abstract algebra. It begins by defining a group as a set with a binary operation that is closed, associative, has an identity element, and where each element has an inverse. It then lists several propositions about properties of groups, including that a group has a unique identity and each element has a unique inverse. The document continues defining additional algebraic structures like rings, fields, subgroups, and properties of groups like cyclic groups. It concludes by discussing matrix groups and their properties.
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 summarizes key results about the structure of the unit group of a group ring R(G,K) where G is a finite Abelian group and K is the integer ring of a finite algebraic extension of the rational field.
It shows that R(G,K) decomposes as a direct sum of fields, each isomorphic to an extension of K. It determines a basis for R(G,K) and describes the structure of the unit group of its integer ring. It also proves that elements of finite order in the unit group of R(G,C) are trivial, and the ranks of this unit group and the integer ring unit group are equal.
This document summarizes research on contraction mappings and their generalizations. It defines key terms like fixed point and contraction mapping. It presents Banach's fixed point theorem, which states that every contraction mapping on a complete metric space has a unique fixed point. It proves this theorem and shows that the iterated sequence of a contraction mapping converges to the unique fixed point. The document also discusses compatible mappings and presents proofs about compatible pairs of mappings being of type (T) or (I). It acknowledges the contributions of others in the field.
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
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.
Normal subgroups are subgroups where conjugation does not affect membership. A subgroup N of a group G is normal if gng-1 is in N for all g in G and n in N. A subgroup N is normal if and only if every left coset of N is also a right coset of N. If every left coset equals a right coset, then conjugation preserves membership in N, making N a normal subgroup.
This document summarizes a research paper on chaotic group actions. It begins with an abstract discussing how group actions can be considered "chaotic" if they exhibit sensitive dependence on initial conditions and have a dense set of points with finite orbits. The paper then provides definitions and examples of chaotic group actions, discussing how the concept generalizes the definition of chaotic maps. It explores which groups can admit chaotic dynamics on topological spaces and which spaces admit chaotic group actions. Specifically, it shows a group has a chaotic action if and only if it is residually finite. It also constructs examples of chaotic actions and proves several theorems about when group actions are chaotic.
The document provides an overview of modular arithmetic and its applications to finding square roots in modular arithmetic. It defines congruences and properties of modular arithmetic. It discusses cyclic groups and their relationship to integers and modular addition/multiplication. It introduces concepts like the order of an element, Lagrange's theorem, and Sylow theorems. It also defines quadratic residues, Legendre symbols, and provides an example of finding a square root in a finite field.
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 summarizes a research paper on defining and analyzing the mass of asymptotically hyperbolic manifolds. The paper establishes that the total mass can be defined unambiguously for such manifolds. It proves a positive mass theorem using spinor methods under certain conditions on the scalar curvature. The paper also analyzes how the total mass is related to the geometry of the manifold near infinity and establishes that the total mass is invariant under changes of coordinates. It proposes generalizing the definition of Hawking mass to asymptotically hyperbolic manifolds and relates the total mass to the Hawking mass of outermost surfaces.
Suborbits and suborbital graphs of the symmetric group acting on ordered r ...Alexander Decker
]
This document discusses suborbits and suborbital graphs of the symmetric group Sn acting on the set X of all ordered r-element subsets from a set X of size n. Some key points:
- Suborbits are orbits of the stabilizer Gx of a point x under the group action. Suborbital graphs are constructed from suborbits.
- Theorems characterize when a suborbit is self-paired or when two suborbits are paired in terms of properties of the permutations that define the suborbits.
- Formulas are derived for the number of self-paired suborbits in terms of the cycle structure of group elements and character values.
El documento habla sobre las competencias disciplinares y su importancia en la formación de los estudiantes. Se dividen en dos niveles: disciplinares básicas y disciplinares extendidas. El objetivo es que los estudiantes desarrollen procesos mentales complejos para enfrentar diversas situaciones a lo largo de su vida, más allá de sólo memorizar conocimientos.
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.
The document provides notes on group theory. It discusses the definition of groups and examples of groups such as (Z, +), (Q, ×), and Sn. Properties of groups like Lagrange's theorem and criteria for subgroups are also covered. The notes then discuss symmetry groups, defining isometries of R2 and showing that the set of isometries forms a group. Symmetry groups G(Π) of objects Π in R2 are introduced and shown to be subgroups. Specific examples of symmetry groups like those of triangles, squares, regular n-gons, and infinite strips are analyzed. Finally, the concept of group isomorphism is defined and examples are given to illustrate isomorphic groups.
IRJET - Triple Factorization of Non-Abelian Groups by Two Minimal SubgroupsIRJET Journal
This document presents research on the triple factorization of non-abelian groups by two minimal subgroups. It begins with an abstract discussing the triple factorization of a group G as G = ABA, where A and B are proper subgroups of G. It then studies the triple factorizations of two classes of non-abelian finite groups: the dihedral groups D2n and the projective special linear groups PSL(2,2n). Several lemmas and a main theorem are presented regarding the triple factorizations and related rank-two coset geometries of these groups. In particular, the theorem discusses conditions for the existence of non-degenerate triple factorizations of D2n and properties of the associated rank-two coset
On The Properties of Finite Nonabelian Groups with Perfect Square Roots Using...inventionjournals
In this paper we determine some internal properties of non abelian groups where the centre Z(G) takes its maximum size. With this restriction we discover that the group G is necessarily a group with perfect square root as shown in our results. We also show that the property of Z(G) is inherited by direct product of groups. Furthermore groups whose centre is as large as possible were constructed.
BCA_Semester-II-Discrete Mathematics_unit-i Group theoryRai University
This document provides an introduction to group theory, including definitions of key concepts such as binary operations, groups, abelian groups, subgroups, cyclic groups, and permutation groups. It defines what constitutes a group and subgroup. Theorems covered include Lagrange's theorem about the order of subgroups dividing the group order, and that every subgroup of a cyclic group is cyclic. Examples are provided of groups defined by binary operations and permutation groups. Exercises at the end involve applying the concepts to specific groups and proving properties of groups.
This document provides definitions and propositions related to abstract algebra. It begins by defining a group as a set with a binary operation that is closed, associative, has an identity element, and where each element has an inverse. It then lists several propositions about properties of groups, including that a group has a unique identity and each element has a unique inverse. The document continues defining additional algebraic structures like rings, fields, subgroups, and properties of groups like cyclic groups. It concludes by discussing matrix groups and their properties.
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 summarizes key results about the structure of the unit group of a group ring R(G,K) where G is a finite Abelian group and K is the integer ring of a finite algebraic extension of the rational field.
It shows that R(G,K) decomposes as a direct sum of fields, each isomorphic to an extension of K. It determines a basis for R(G,K) and describes the structure of the unit group of its integer ring. It also proves that elements of finite order in the unit group of R(G,C) are trivial, and the ranks of this unit group and the integer ring unit group are equal.
This document summarizes research on contraction mappings and their generalizations. It defines key terms like fixed point and contraction mapping. It presents Banach's fixed point theorem, which states that every contraction mapping on a complete metric space has a unique fixed point. It proves this theorem and shows that the iterated sequence of a contraction mapping converges to the unique fixed point. The document also discusses compatible mappings and presents proofs about compatible pairs of mappings being of type (T) or (I). It acknowledges the contributions of others in the field.
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
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.
Normal subgroups are subgroups where conjugation does not affect membership. A subgroup N of a group G is normal if gng-1 is in N for all g in G and n in N. A subgroup N is normal if and only if every left coset of N is also a right coset of N. If every left coset equals a right coset, then conjugation preserves membership in N, making N a normal subgroup.
This document summarizes a research paper on chaotic group actions. It begins with an abstract discussing how group actions can be considered "chaotic" if they exhibit sensitive dependence on initial conditions and have a dense set of points with finite orbits. The paper then provides definitions and examples of chaotic group actions, discussing how the concept generalizes the definition of chaotic maps. It explores which groups can admit chaotic dynamics on topological spaces and which spaces admit chaotic group actions. Specifically, it shows a group has a chaotic action if and only if it is residually finite. It also constructs examples of chaotic actions and proves several theorems about when group actions are chaotic.
The document provides an overview of modular arithmetic and its applications to finding square roots in modular arithmetic. It defines congruences and properties of modular arithmetic. It discusses cyclic groups and their relationship to integers and modular addition/multiplication. It introduces concepts like the order of an element, Lagrange's theorem, and Sylow theorems. It also defines quadratic residues, Legendre symbols, and provides an example of finding a square root in a finite field.
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 summarizes a research paper on defining and analyzing the mass of asymptotically hyperbolic manifolds. The paper establishes that the total mass can be defined unambiguously for such manifolds. It proves a positive mass theorem using spinor methods under certain conditions on the scalar curvature. The paper also analyzes how the total mass is related to the geometry of the manifold near infinity and establishes that the total mass is invariant under changes of coordinates. It proposes generalizing the definition of Hawking mass to asymptotically hyperbolic manifolds and relates the total mass to the Hawking mass of outermost surfaces.
Suborbits and suborbital graphs of the symmetric group acting on ordered r ...Alexander Decker
]
This document discusses suborbits and suborbital graphs of the symmetric group Sn acting on the set X of all ordered r-element subsets from a set X of size n. Some key points:
- Suborbits are orbits of the stabilizer Gx of a point x under the group action. Suborbital graphs are constructed from suborbits.
- Theorems characterize when a suborbit is self-paired or when two suborbits are paired in terms of properties of the permutations that define the suborbits.
- Formulas are derived for the number of self-paired suborbits in terms of the cycle structure of group elements and character values.
El documento habla sobre las competencias disciplinares y su importancia en la formación de los estudiantes. Se dividen en dos niveles: disciplinares básicas y disciplinares extendidas. El objetivo es que los estudiantes desarrollen procesos mentales complejos para enfrentar diversas situaciones a lo largo de su vida, más allá de sólo memorizar conocimientos.
Este documento presenta un laboratorio sobre regresiones logísticas realizado en SPSS. Explica cómo ingresar y seleccionar variables, ejecutar un análisis de regresión logística binaria en SPSS e interpretar los resultados. Concluye que SPSS es un programa útil para análisis estadísticos y la gestión de datos, y recomienda aprender a usarlo para trabajos estadísticos.
Este documento describe cómo realizar un análisis de regresión lineal múltiple en SPSS. Explica los objetivos del análisis de regresión, los pasos para ingresar y seleccionar datos en SPSS, y cómo interpretar los resultados. También incluye conclusiones sobre la utilidad de SPSS para análisis estadísticos y recomendaciones para aprender a usar este programa.
Emily Marvin is a senior at Hatboro-Horsham High School who is anticipated to graduate in June 2012. She has taken several honors and AP courses including AP Biology, AP Statistics, and AP Calculus. Emily has work experience as a counter worker and flower maintenance employee at Valley Green Flowers & Gift shop and has babysat young children. She has received awards for her involvement in basketball and is involved in many extracurricular activities including Future Business Leaders of America, Student Council, and the girls basketball team.
Una computadora está compuesta por hardware y software. El hardware incluye componentes físicos como el monitor, teclado y procesador, mientras que el software son programas como el sistema operativo y aplicaciones. El hardware se compone de la CPU, memoria y periféricos, y el software incluye sistemas operativos, programas de aplicación y desarrollo. La computadora funciona mediante la interacción del usuario con el sistema operativo a través de la CPU y la memoria RAM para ejecutar programas almacenados en el disco duro.
Este documento presenta un capítulo sobre la organización y arquitectura de computadores. El capítulo es escrito por el Ing. Luis Fernando Aguas B. de la Facultad de Ingeniería, Escuela de Sistemas de la universidad.
Presentación sobre la sesión "Big Data: the Management Revolution", dentro del Programa Ejecutivo de Big Data y Business Intelligence celebrado en Madrid en Febrero de 2016, en nuestra sede de la Universidad de Deusto.
Este documento trata sobre los elementos básicos del lenguaje de programación C#. Explica las clases, objetos, tipos de datos, interfaces, estructuras, propiedades, eventos y otros conceptos fundamentales del lenguaje. También describe las herramientas disponibles para desarrollar en C#, como el compilador C# y el entorno de desarrollo Visual Studio.
The document discusses groups and graphs in probability theory. It provides background on the probability that a group element fixes a set and related work applying this concept to graph theory, specifically generalized conjugacy class graphs and orbit graphs. The main results are:
1) For a dihedral group of order 2n acting regularly on the set of commuting pairs of elements, the probability that a group element fixes a set is 1/n.
2) Under this action, the generalized conjugacy class graph of the dihedral group has n isolated vertices and is empty.
3) The orbit graph vertices correspond to the n orbits of sizes 1 or 2n, with an edge between vertices if their corresponding orbits are conjugate.
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 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.
Brian Covello: Research in Mathematical Group Representation Theory and SymmetryBrian Covello
Brian Covello's research review on group representation theory and symmetry. In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication. Representations of groups are important because they allow many group-theoretic problems to be reduced to problems in linear algebra, which is well understood. They are also important in physics because, for example, they describe how the symmetry group of a physical system affects the solutions of equations describing that system.
The term representation of a group is also used in a more general sense to mean any "description" of a group as a group of transformations of some mathematical object. More formally, a "representation" means a homomorphism from the group to the automorphism group of an object. If the object is a vector space we have a linear representation. Some people use realization for the general notion and reserve the term representation for the special case of linear representations.
In mathematics, representation theory is a technique for analyzing abstract groups in terms of groups of linear transformations.
For the symmetric groups, a graphical method exists to determine their finite representations that associates with each representation a Young tableau (also known as a Young diagram). The direct product of two representations may easily be decomposed into a direct sum of irreducible representation by a set of rules for the "direct product" of two Young diagrams. Each diagram also contains information about the dimension of the representation to which it corresponds. Young tableaux provide a far cleaner way of working with representations than the algebraic methods that underlie their use.
This is a journal concise version (without diagrams and figures) of the preprint arXiv:1308.4060.
Abstract: Polyadic systems and their representations are reviewed and a classification of general polyadic systems is presented. A new multiplace generalization of associativity preserving homomorphisms, a 'heteromorphism' which connects polyadic systems having unequal arities, is introduced via an explicit formula, together with related definitions for multiplace representations and multiactions. Concrete examples of matrix representations for some ternary groups are then reviewed. Ternary algebras and Hopf algebras are defined, and their properties are studied. At the end some ternary generalizations of quantum groups and the Yang-Baxter equation are presented.
A NEW APPROACH TO M(G)-GROUP SOFT UNION ACTION AND ITS APPLICATIONS TO M(G)-G...ijscmcj
In this paper, we define a new type of M(G)-group action , called M(G)-group soft union(SU) action and M(G)-ideal soft union(SU) action on a soft set. This new concept illustrates how a soft set effects on an M(G)-group in the mean of union and inclusion of sets and its function as bridge among soft set theory, set theory and M(G)-group theory. We also obtain some analog of classical M(G)- group theoretic concepts for M(G)-group SU-action. Finally, we give the application of SU-actions on M(G)-group to M(G)-group theory.
A Complete Solution Of Markov S Problem On Connected Group TopologiesLisa Muthukumar
This document provides a solution to Markov's problem regarding connected group topologies for abelian groups. The key points are:
1. The authors prove that every abelian M-group (a group where all proper unconditionally closed subgroups have index at least c) admits a densely pathwise connected and locally densely pathwise connected group topology.
2. This completes the solution to Markov's problem, as it shows that an abelian group admits a connected group topology if and only if it is an M-group.
3. The proof involves characterizing w-divisible groups (groups where all subgroups have the same cardinality) and using a classical construction to put connected topologies on homogeneous groups
This document presents definitions and results regarding L-fuzzy normal sub l-groups. It begins with introductions and preliminaries on L-fuzzy sets, L-fuzzy subgroups, and L-fuzzy sub l-groups. It then presents 8 theorems on properties of L-fuzzy normal sub l-groups, such as conditions for an L-fuzzy subset to be an L-fuzzy normal sub l-group, the intersection of L-fuzzy normal sub l-groups also being an L-fuzzy normal sub l-group, and conditions where an L-fuzzy sub l-group is necessarily an L-fuzzy normal sub l-group. The document references 6 sources and is focused on developing the theory of L-fuzzy
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2. 900 Mustafa Anis El-Sanfaz
The first investigation of the commutativity degree of symmetric groups was done
in 1968 by Erdos and Turan [2]. Few years later, Gustafson [3] and MacHale [4]
found an upper bound for the commutativity degree of all finite non-abelian groups in
which
5
.
8
P G The concept of the commutativity degree has been generalized and
extended by several authors. In 1975, a new concept was introduced by Sherman [5],
namely the probability of an automorphism of a finite group fixes an arbitrary element
in the group. The definition of this probability is given as follows:
Definition 1.2
[5] Let G be a group. Let X be a non-empty set of G (G is a group of permutations
of ).X Then the probability of an automorphism of a group fixes a random element
from X is defined as follows:
, , ,
.G
g x gx x g G x X
P X
G X
In 2011, Moghaddam [6] explored Sherman's definition and introduced a new
probability, which is called the probability of an automorphism fixes a subgroup
element of a finite group, the probability is stated as follows:
, | : ,
, ,G
G
A
h h h H A
P H G
H G
where GA is the group of automorphisms of a group .G It is clear that if ,H G then
, .G GA AP G G P
Recently, Omer et al [7] extended the work in [5] by defining the probability that a
group element fixes a set. In this paper, the probability that a group element fixes a set
is found for metacyclic 2-groups of positive type of nilpotency class at least three.
Next, we state some basic concepts in group theory that are needed in this paper.
Definition 1.3
[8] Suppose G is a finite group that acts on a set .S A group G acts regularly on S
if the action is transitive and Stab 1G s for all .s S Also, the action is regular if
for 1s and 2s belong to ,S there exists an element g G such that 1 2.gs s
Definition 1.4
[9] Let G acts on a set S and .x S The orbit of x denoted by cl x is the sub set
cl : .x gx g G S
Next, we provide some concepts related to metacyclic p-groups. Throughout this
paper, p denotes a prime number.
3. On The Probability That An Element of Metacyclic 2-Group of Positive et.al. 901
Definition 1.5
[10] A group G is called a metacyclic if it has a cyclic normal subgroup H such that
the quotient group G
H
is also cyclic.
In 1973, King [11] gave the presentation of metacyclic p-groups, as given in the
follows:
1 2
1 1
1 2, : 1, , , , where , 0, 0, , | 1 .p p n m
G a b a b a b a a m n p p n m
In 2005, Beuerle [12] separated the classification into two parts, namely for the
non-abelian metacyclic p-groups of class two and class at least three. Based on [12],
the metacyclic p- groups of nilpotency class two are then partitioned into two families
of non-isomorphic p-groups stated as follows:
8.
, : 1, 1, , ,where , , , 2 and 1.p p p
G a b a b a b a
G Q
Meanwhile, the metacyclic p-groups of nilpotency class of at least three (p is an
odd prime) are partitioned into the following groups:
, : 1, 1, , , where , , , 1 2 and .
, : 1, , , , where , , , , 1 2 ,
and .
p p p
p p p p
G a b a b b a a
G a b a b a b a a
Moreover, metacyclic p-groups are also classified into two types, namely negative
and positive type [12]. The following notations for these two types which are used in
this paper are represented as follows:
, , , , , : 1, , , , where , , , , 1.p p p t
G a b a b a b a a t p
If 1t p , then the group is called a metacyclic of negative type and it is of
positive type if 1t p . Thus, , , , ,G denotes the metacyclic group of
negative type, while , , , ,G denotes the positive type. These two notations
are shortened to ,G p and , ,G p respectively ([11], [12]).
Since this paper focuses on metacyclic p-groups of positive type, thus the
following theorem gives the presentations of the two groups of positive type.
Theorem 1.1
[12] Let G be a metacyclic 2-group of positive type of nilpotency class of at least
three. Then G is isomorphic to one of the following types:
2 2 2
1.1.1 , : 1, , , , , , 1 2 .G a b a b b a a
2 2 2 2
1.1.2 , : 1, , , ,1 2 , , .G a b a b a b a a
4. 902 Mustafa Anis El-Sanfaz
The followings are some basic concepts of graph theory that are needed in this
paper. These concepts can be found in one of the references ([13], [14]).
A graph is a mathematical structure consisting of two sets namely vertices and
edges which are denoted by V and ,E respectively. The graph is called
directed if its edges are identified with ordered pair of vertices. Otherwise, is called
indirected. Two vertices are adjacent if they are linked by an edge. A complete graph
is a graph where each ordered pair of distinct n vertices are adjacent, denoted by ,nK
while it is null if there is no vertices in . The graph is called empty if there are no
edges linked between its vertices. In this paper, eK denotes empty graph, while 0K
denotes null graph.
In 1990, a new graph called graph related to conjugacy class was introduced by
Bertram et al. [15]. The vertices of this graph are non central conjugacy classes
i.e ,V K G Z G where K G is the number of conjugacy class of a group
and Z G is the center of a group .G A pair of vertices of this graph are connected by
an edge if their cardinalities are not coprime.
In 2013, Omer et al. [16] extended the work on the conjugacy class graph by
introducing a generalized conjugacy class graph whose vertices are non-central orbits
under group action on a set.
This paper is structured as follows: Section 1 provides some fundamental concepts
of group theory and graph theory which are used in this paper. In Section 2, we state
some of previous works, which are related to the commutativity degree, in particular
related to the probability that a group element fixes a set and conjugacy class graph.
The main results are presented in Section 3.
Preliminaries
In this section, we provide some previous works related to the commutativity degree,
more precisely to the probability that an element of a group fixes a set and graph
theory, in particular to graph related to conjugacy class.
Recently, Omer et al. [7] extended the commutativity degree by defining the
probability that an element of a group fixes a set of size two, given in the following.
Definition 2.1
[7] Let G be a group. Let S be a set of all subsets of commuting elements of G of
size two. If G acts on ,S then the probability of an element of a group fixes a set is
defined as follows:
, | : ,
.G
g s gs s g G s S
P S
S G
5. On The Probability That An Element of Metacyclic 2-Group of Positive et.al. 903
Theorem 2.1
[7] Let G be a finite group and let X be a set of elements of G of size two in the
form of ,a b where a and b commute. Let S be the set of all subsets of commuting
elements of G of size two and G acts on .S Then the probability that an element of a
group fixes a set is given by
,G
K S
P S
S
where K S is the number of orbits of S in .G
In 2014, Mustafa et al. [17] has extended the work in [7] by restricting the order of
. The following theorem illustrates their results.
Theorem 2.2
[17] Let G be a finite group and let S be a set of elements of G of size two in the
form of , ,a b where ,a b commute and 2.a b Let be the set of all subsets of
commuting elements of G of size two and G acts on .Then the probability that an
element of a group fixes a set is given by ,G
K
P where K is the
number of conjugacy classes of in .G
Recently, Mustafa et al. [18] obtained the probability that an element of a group
fixes a set for metacyclic 2-groups of negative type of nilpotency class two and class
at least three. The probability is also computed under conjugate action for metacyclic
2-groups of positive type of nilpotency class at least three [19]. In addition, Mustafa et
al. [20] found the probability that a group element fixes a set for semi-dihedral groups
and quasi-dihedral groups.
In this paper, the probability that a positive type metacyclic 2-group element fixes
a set is computed. Furthermore, the obtained results are then applied to graph theory
by using the orbits that are obtained under group action on a set to graph conjugacy
class.
Some related works on conjugacy class graph include Bianchi et al. [21] who
studied the regularity of the graph related to conjugacy classes. In addition, Moreto et
al. [22] in 2005 classified the groups in which conjugacy classes sizes are not coprime
for any five distinct classes. Furthermore, You et al. [23] also classified the groups in
which conjugacy classes are not set-wise relatively prime for any four distinct classes.
Moreover, Moradipour et al. [24] used the graph related to conjugacy classes to find
some graph properties of some finite metacyclic 2-groups.
Recently, Omer et al. [16] introduced generalized conjugacy class graph, given in
the following definition.
Definition 2.2
[16] Let G be a finite group and a set of G. Let A be the set of commuting element
in , i.e : , .g g g G Then the generalized conjugacy class graph c
G is
defined as a graph whose vertices are non-central orbits under group action on a set,
6. 904 Mustafa Anis El-Sanfaz
that is .c
GV K A Two vertices 1 and 2 in c
G are adjacent if their
cardinalities are not coprime, i.e 1 2gcd , 1.
Moreover, the generalized conjugacy class graph is determined for the symmetric
groups and alternating groups [25].
Main Results
This section provides our main results. First, the probability that an element of
metacyclic 2-groups of positive type of nilpotency class at least three fixes a set is
computed. Then the second part relates the obtained results to generalized conjugacy
class graph.
Throughout this section, let S be a set of elements of G of size two in the form of
(a, b), where a and b commute and |a|=|b|=2. Let be the set of all subsets of
commuting elements of G of size two and G acts on by regular action.
The Probability That a Metacyclic 2-Group Element Fixes a Set
Theorem 3.1
Let G be a finite group of type 2 2 2
1.1.1 , , : 1, , ,G a b a b b a a where
, , , 1 2 , . If G acts regularly on , then 1.GP
Proof
If G acts regularly on , then there exists an element g G such that 1 2g for
1 2, . The elements of order two in G are
1 1
2 2
,a b and
1 1
2 2
.a b Therefore,
the elements of are stated as follows: Two elements in the form of
1 1 1
2 2 2
, ,0 2 ,i
a a b i and one element in the form
1 12 1 2
2
, .b a b Then 3
. By regular action, there exists an element g G for 1 2, . such that 1 2.g
The orbits of can be described as follows: One orbit in the form of
1 1
2 2
, ,a b
one orbit in the form of
1 1 1
2 2 2
,a a b and one orbit in the form of
1 12 1 2
2
, .b a b
Thus there are three orbits. Using Theorem 2.2, then 1.GP
Theorem 3.2
Let G be group of type 2
1.1.2 , , : 1,G a b a 2
b 2 2
, ,,a b a a
1 2 , , . If G acts regularly on , then 1.GP
7. On The Probability That An Element of Metacyclic 2-Group of Positive et.al. 905
Proof
If G acts regularly on , then there exists an element g G such that 1 2g for
1 2, .
4 1 1
7 2 2 2
. Thus, the elements of order two in are ,g G G a b
4 1
7 2 2
a b
and
1
2
.a Hence, the elements of are stated as follows: Two elements are in the
form of
1 4 1 1
2 7 2 2 2
( , ),0 1,i
a a b i and there is one element in which are in the
form of
4 1 4 1 1
7 2 2 7 2 2 2
of ( , ),a b a b from which it follows that 3. By regular
action, there exists an element g G for 1 2, . such that 1 2.g The orbits of
can be described as follows: One orbit in the form of
1 4 1
2 7 2 2
( , ),a a b one orbit
in the form of
1 4 1 1
2 7 2 2 2
( , )a a b and one orbit in the form of
4 1 4 1 1
7 2 2 7 2 2 2
( , ).a b a b Thus there are three orbits. Using Theorem 2.2, then
1.GP
The Generalized Conjugacy Class Graph
In this section, the results are related to generalized conjugacy class graph. First, the
generalized conjugacy class graph of metacyclic 2-groups of positive type of
nilpotency class of at least three is found, starting with the group of type (1.1.1).
Theorem 3.3
Let G be a finite group of type 2 2 2
1.1.1 , , : 1, , ,G a b a b b a a where
, , , 1 2 , . If G acts regularly on , then .c
G eK
Proof
Based on Theorem 3.1, there are three orbits all of size one. Using Definition 2.2, c
G
consists of three isolated vertices, which are in the form of
1 12 1 2
2
, ,b a b
1 1 1
2 2 2
,a a b and
1 1
2 2
, .a b The proof then follows. ■
Theorem 3.4
Let G be group of type 2
1.1.2 , , : 1,G a b a 2
b 2 2
, ,,a b a a
1 2 , , . If G acts regularly on , then
0
, if 1,
, if 1.
c e
G
K
K
Proof
In the first case, the proof is similar to the proof of Theorem 3.3. When 1and
based on Theorem 3.3 in [19] 1GP under conjugate action and by using
8. 906 Mustafa Anis El-Sanfaz
Theorem 3.1 in [19] all commuting element of G of order two are in the center of .G
Thus, .K A Using the vertices adjacency of the generalized conjugacy
class graph, c
G is null. ■
Conclusion
In this paper, the probability that a group element fixes a set under regular action is
found for metacyclic 2-groups of positive type of nilpotency class of at least three.
The results obtained are then applied to generalized conjugacy class graph.
Acknowledgments
The first author would like to acknowledge the Ministry of Higher Education in Libya
for his PhD scholarship. The authors would also like to acknowledge Ministry of
Education (MOE) Malaysia and Universiti Teknologi Malaysia for the financial
funding through the Research University Grant (GUP) Vote No 08H07.
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10. 908 Mustafa Anis El-Sanfaz
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