This document summarizes research on algebraic elements in group algebras. It begins by defining a group algebra k[G] over a commutative ring k. An element of k[G] is algebraic if it satisfies a non-constant polynomial. The document discusses tools for studying algebraic elements like partial augmentations corresponding to conjugacy classes. It also summarizes results on idempotents, including Kaplansky's theorem that the trace of an idempotent is real and rational. The author's past work on dimension subgroups is also briefly outlined.
The document discusses normal subgroups and quotient groups. Given a subgroup H that is normal in group G, a quotient group G/H can be constructed whose elements are the cosets of H. If there is an intermediate subgroup K such that H is normal in K and K is a subgroup of G, then K/H is a subgroup of G/H. There is a bijection between the subgroups of G containing a normal subgroup H and the subgroups of the quotient group G/H. An example is given where G = Z, H = pZ for some prime p, and there are only two possible intermediate subgroups K = Z or K = pZ.
The document discusses subgroups of groups, with examples. It defines a subgroup as a subset of a group that is closed under the group's operation and contains inverses and identities. Examples of groups given include the general linear group of invertible matrices, the symmetric group of permutations, and the integers under addition. Subgroups are discussed, including cyclic subgroups generated by a group element and its powers. Specific subgroups of the positive integers under addition are analyzed in detail.
(1) The question asks to find the last two digits of 2^1000. (2) Computing exponents modulo 100, it is shown that 2^20 = 24 (mod 100) and 2^20 = 76 (mod 100). (3) By induction, 2^n = 76 (mod 100) for any exponent n. Therefore, the last two digits of 2^1000 are 76.
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.
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 summarizes research on the energy of graphs. It defines the energy of a graph as the sum of the absolute values of its eigenvalues. It shows that for any positive epsilon, there exist infinitely many values of n for which a k-regular graph of order n exists whose energy is arbitrarily close to the known upper bound for k-regular graphs. It also establishes the existence of equienergetic graphs that are not cospectral. Equienergetic graphs have the same energy even if they do not have the same spectrum of eigenvalues.
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.
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.
The document discusses normal subgroups and quotient groups. Given a subgroup H that is normal in group G, a quotient group G/H can be constructed whose elements are the cosets of H. If there is an intermediate subgroup K such that H is normal in K and K is a subgroup of G, then K/H is a subgroup of G/H. There is a bijection between the subgroups of G containing a normal subgroup H and the subgroups of the quotient group G/H. An example is given where G = Z, H = pZ for some prime p, and there are only two possible intermediate subgroups K = Z or K = pZ.
The document discusses subgroups of groups, with examples. It defines a subgroup as a subset of a group that is closed under the group's operation and contains inverses and identities. Examples of groups given include the general linear group of invertible matrices, the symmetric group of permutations, and the integers under addition. Subgroups are discussed, including cyclic subgroups generated by a group element and its powers. Specific subgroups of the positive integers under addition are analyzed in detail.
(1) The question asks to find the last two digits of 2^1000. (2) Computing exponents modulo 100, it is shown that 2^20 = 24 (mod 100) and 2^20 = 76 (mod 100). (3) By induction, 2^n = 76 (mod 100) for any exponent n. Therefore, the last two digits of 2^1000 are 76.
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.
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 summarizes research on the energy of graphs. It defines the energy of a graph as the sum of the absolute values of its eigenvalues. It shows that for any positive epsilon, there exist infinitely many values of n for which a k-regular graph of order n exists whose energy is arbitrarily close to the known upper bound for k-regular graphs. It also establishes the existence of equienergetic graphs that are not cospectral. Equienergetic graphs have the same energy even if they do not have the same spectrum of eigenvalues.
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.
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.
Total Dominating Color Transversal Number of Graphs And Graph Operationsinventionjournals
Total Dominating Color Transversal Set of a graph is a Total Dominating Set of the graph which is also Transversal of Some 휒 - Partition of the graph. Here 휒 is the Chromatic number of the graph. Total Dominating Color Transversal number of a graph is the cardinality of a Total Dominating Color Transversal Set which has minimum cardinality among all such sets that the graph admits. In this paper, we consider the well known graph operations Join, Corona, Strong product and Lexicographic product of graphs and determine Total Dominating Color Transversal number of the resultant graphs.
A crystallographic group is a group acting on R^n that contains a translation subgroup Z^n as a finite index subgroup. Here we consider which Coxeter groups are crystallographic groups. We also expose the enumeration in dimension 2 and 3. Then we shortly give the principle under which the enumeration of N dimensional crystallographic groups is done.
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 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.
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.
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 provides solutions to problems in group theory from the book Topics in Algebra by I.N. Herstein. The solutions cover problems related to determining if a system forms a group, properties of groups like abelian groups, and examples in the symmetric group S3. The preface explains that the solutions are meant to facilitate deeper understanding and some notations were changed for clarity.
Applications and Properties of Unique Coloring of GraphsIJERA Editor
This paper studies the concepts of origin of uniquely colorable graphs, general results about unique vertex colorings, assorted results about uniquely colorable graphs, complexity results for unique coloring Mathematics Subject Classification 2000: 05CXX, 05C15, 05C20, 37E25.
A common fixed point of integral type contraction in generalized metric spacessAlexander Decker
This document presents a common fixed point theorem for two self-mappings S and T on a G-metric space X that satisfies a contractive condition of integral type. It begins with definitions related to G-metric spaces and contractive conditions. It then states Theorem 1.1, which proves that if S and T satisfy the given integral type contractive condition, along with other listed conditions, then S and T have a unique point of coincidence in X. If S and T are also weakly compatible, then they have a unique common fixed point. The proof of Theorem 1.1 is then provided.
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.
This document summarizes the work of Raffaele Rainone on deriving bounds on the dimension of fixed point spaces for actions of classical algebraic groups. It begins by introducing algebraic groups and their actions on varieties. It then discusses conjugacy classes and computing dimensions of centralizers for elements of classical groups. The main results provide global and local bounds on the dimension of fixed point spaces for elements of prime order when the group is a classical group and the variety consists of cosets for certain geometric subgroups. Several open problems are posed regarding improving these bounds.
A Szemeredi-type theorem for subsets of the unit cubeVjekoslavKovac1
This document summarizes a talk on gaps between arithmetic progressions in subsets of the unit cube. It presents three key propositions:
1) For subsets A of positive measure, structured progressions contribute a lower bound depending on the measure of A and the best known bounds for Szemerédi's theorem.
2) Estimating errors by pigeonholing scales, the difference between smooth and sharp progressions over various scales is bounded above by a sublinear function of scales.
3) For sufficiently nice subsets, the difference between measure and smoothed measure is arbitrarily small by choosing a small smoothing parameter.
Combining these propositions shows that for sufficiently nice subsets, gaps between progressions contain an interval
1. The document discusses numerical calculations of Darmon points, which are conjectural algebraic points on elliptic curves defined over real quadratic fields.
2. Darmon points are constructed using p-adic integration on the upper half-plane and cohomology classes associated to the elliptic curve.
3. Examples of calculating Darmon points are presented for various elliptic curves and real quadratic fields using Sage code implementing the necessary algorithms.
Este documento trata sobre el módulo 4 de un curso de educación por competencias. El módulo se enfoca en competencias y contenidos procedimentales. Incluye temas sobre cómo aprender a usar el conocimiento, facilitar el aprendizaje de contenidos procedimentales, y la relación entre procedimientos, currículo y evaluación. También presenta un ejemplo sobre cómo una maestra enseña a sus estudiantes a preparar un postre mexicano sin darles instrucciones específicas.
Este documento resume los orígenes y fundamentos de la terapia Gestalt. Nace en Alemania en 1912 de los estudios de Max Wertheimer, Kurt Koffka y Wolfang Kohler sobre la percepción, que ven al organismo y el medio como un todo significativo estructurado en figura y fondo. Más tarde, Fritz y Laura Perls desarrollan la psicoterapia Gestalt, enfocada en el aquí y ahora y basada en una visión fenomenológica e humanista que ve a la Gestalt como el estudio de la totalidad y forma del organismo human
The beginning section of my novella, The Courtesans of Abaddon, in which four sisters work in a brothel that exists simultaneously within heaven and hell, serving both angels and demons, in which time both passes eternally and doesn't exist at all, where day and night, light and dark, good and evil, god and anti-god, seem to be one and the same.
1) The document discusses the key findings of a survey of 150 CIOs and CTOs across the UK, France, and Germany on their companies' use of cloud computing.
2) The main drivers for cloud adoption were reported to be lower operating costs (30%) and lower infrastructure costs (30%). Nearly all respondents used public cloud and most implemented cloud differently across departments.
3) Sales/CRM (30%) and marketing (19%) were the most common departments for cloud implementation. The CIO/CTO was typically the main decision maker for cloud (49%).
Kami, PT. Green Global Indonesia , sebagai satu perusahaan event organizer mampu memberikan layanan One Stop Service untuk memenuhi semua kebutuhan pelanggannya.
The document discusses group rings and zero divisors in group rings. Some key points:
- A group ring K[G] is a vector space over a field K with basis G, where elements are finite formal sums of terms with coefficients in K. Multiplication is defined distributively using the group multiplication in G.
- If G contains an element of finite order, then K[G] will contain zero divisors. However, if G has no elements of finite order, it is not clear if K[G] contains zero divisors.
- The document proves a theorem: K[G] is prime (has no zero divisors) if and only if G has no non-identity finite normal subgroups.
Total Dominating Color Transversal Number of Graphs And Graph Operationsinventionjournals
Total Dominating Color Transversal Set of a graph is a Total Dominating Set of the graph which is also Transversal of Some 휒 - Partition of the graph. Here 휒 is the Chromatic number of the graph. Total Dominating Color Transversal number of a graph is the cardinality of a Total Dominating Color Transversal Set which has minimum cardinality among all such sets that the graph admits. In this paper, we consider the well known graph operations Join, Corona, Strong product and Lexicographic product of graphs and determine Total Dominating Color Transversal number of the resultant graphs.
A crystallographic group is a group acting on R^n that contains a translation subgroup Z^n as a finite index subgroup. Here we consider which Coxeter groups are crystallographic groups. We also expose the enumeration in dimension 2 and 3. Then we shortly give the principle under which the enumeration of N dimensional crystallographic groups is done.
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 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.
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.
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 provides solutions to problems in group theory from the book Topics in Algebra by I.N. Herstein. The solutions cover problems related to determining if a system forms a group, properties of groups like abelian groups, and examples in the symmetric group S3. The preface explains that the solutions are meant to facilitate deeper understanding and some notations were changed for clarity.
Applications and Properties of Unique Coloring of GraphsIJERA Editor
This paper studies the concepts of origin of uniquely colorable graphs, general results about unique vertex colorings, assorted results about uniquely colorable graphs, complexity results for unique coloring Mathematics Subject Classification 2000: 05CXX, 05C15, 05C20, 37E25.
A common fixed point of integral type contraction in generalized metric spacessAlexander Decker
This document presents a common fixed point theorem for two self-mappings S and T on a G-metric space X that satisfies a contractive condition of integral type. It begins with definitions related to G-metric spaces and contractive conditions. It then states Theorem 1.1, which proves that if S and T satisfy the given integral type contractive condition, along with other listed conditions, then S and T have a unique point of coincidence in X. If S and T are also weakly compatible, then they have a unique common fixed point. The proof of Theorem 1.1 is then provided.
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.
This document summarizes the work of Raffaele Rainone on deriving bounds on the dimension of fixed point spaces for actions of classical algebraic groups. It begins by introducing algebraic groups and their actions on varieties. It then discusses conjugacy classes and computing dimensions of centralizers for elements of classical groups. The main results provide global and local bounds on the dimension of fixed point spaces for elements of prime order when the group is a classical group and the variety consists of cosets for certain geometric subgroups. Several open problems are posed regarding improving these bounds.
A Szemeredi-type theorem for subsets of the unit cubeVjekoslavKovac1
This document summarizes a talk on gaps between arithmetic progressions in subsets of the unit cube. It presents three key propositions:
1) For subsets A of positive measure, structured progressions contribute a lower bound depending on the measure of A and the best known bounds for Szemerédi's theorem.
2) Estimating errors by pigeonholing scales, the difference between smooth and sharp progressions over various scales is bounded above by a sublinear function of scales.
3) For sufficiently nice subsets, the difference between measure and smoothed measure is arbitrarily small by choosing a small smoothing parameter.
Combining these propositions shows that for sufficiently nice subsets, gaps between progressions contain an interval
1. The document discusses numerical calculations of Darmon points, which are conjectural algebraic points on elliptic curves defined over real quadratic fields.
2. Darmon points are constructed using p-adic integration on the upper half-plane and cohomology classes associated to the elliptic curve.
3. Examples of calculating Darmon points are presented for various elliptic curves and real quadratic fields using Sage code implementing the necessary algorithms.
Este documento trata sobre el módulo 4 de un curso de educación por competencias. El módulo se enfoca en competencias y contenidos procedimentales. Incluye temas sobre cómo aprender a usar el conocimiento, facilitar el aprendizaje de contenidos procedimentales, y la relación entre procedimientos, currículo y evaluación. También presenta un ejemplo sobre cómo una maestra enseña a sus estudiantes a preparar un postre mexicano sin darles instrucciones específicas.
Este documento resume los orígenes y fundamentos de la terapia Gestalt. Nace en Alemania en 1912 de los estudios de Max Wertheimer, Kurt Koffka y Wolfang Kohler sobre la percepción, que ven al organismo y el medio como un todo significativo estructurado en figura y fondo. Más tarde, Fritz y Laura Perls desarrollan la psicoterapia Gestalt, enfocada en el aquí y ahora y basada en una visión fenomenológica e humanista que ve a la Gestalt como el estudio de la totalidad y forma del organismo human
The beginning section of my novella, The Courtesans of Abaddon, in which four sisters work in a brothel that exists simultaneously within heaven and hell, serving both angels and demons, in which time both passes eternally and doesn't exist at all, where day and night, light and dark, good and evil, god and anti-god, seem to be one and the same.
1) The document discusses the key findings of a survey of 150 CIOs and CTOs across the UK, France, and Germany on their companies' use of cloud computing.
2) The main drivers for cloud adoption were reported to be lower operating costs (30%) and lower infrastructure costs (30%). Nearly all respondents used public cloud and most implemented cloud differently across departments.
3) Sales/CRM (30%) and marketing (19%) were the most common departments for cloud implementation. The CIO/CTO was typically the main decision maker for cloud (49%).
Kami, PT. Green Global Indonesia , sebagai satu perusahaan event organizer mampu memberikan layanan One Stop Service untuk memenuhi semua kebutuhan pelanggannya.
The document discusses group rings and zero divisors in group rings. Some key points:
- A group ring K[G] is a vector space over a field K with basis G, where elements are finite formal sums of terms with coefficients in K. Multiplication is defined distributively using the group multiplication in G.
- If G contains an element of finite order, then K[G] will contain zero divisors. However, if G has no elements of finite order, it is not clear if K[G] contains zero divisors.
- The document proves a theorem: K[G] is prime (has no zero divisors) if and only if G has no non-identity finite normal subgroups.
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.
cryptography slides. it consists of all the lecture notes of ankur sodhi. students of lpu final year btech computer sc. can take it as a reference if needed
1. The document introduces algebraic structures that are important in cryptography such as groups, rings, and fields.
2. A group is a set of elements with an operation that is associative, has an identity element, and has inverses. Examples of groups include the integers modulo n under addition.
3. Fields require all the properties of groups and rings, plus every non-zero element must have a multiplicative inverse. Finite fields, called Galois fields, play a key role in cryptography.
1. The document introduces algebraic structures that are important in cryptography such as groups, rings, and fields.
2. A group is a set of elements with an operation that is associative, has an identity element, and has inverses. Examples of groups include the integers modulo n under addition.
3. Finite fields play a key role in cryptography and consist of a finite number of elements that form a field. They are often denoted as GF(p) or GF(2n).
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.
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.
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
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.
1. Representation theory studies how algebraic structures like groups, algebras, and Lie algebras can be represented by linear transformations on vector spaces. Quiver representations assign vector spaces to vertices and linear maps to arrows of a quiver.
2. Hall algebras were introduced to study representations of quivers. The document outlines representation theory, quivers, Hall algebras, and connections between quivers, Lie theory, and quantum groups.
3. Representation theory has applications in many areas of mathematics including algebra, analysis, algebraic geometry, and topology. Dynkin diagrams classify semisimple Lie algebras and Kac-Moody algebras. Quantum groups are quantized enveloping algebras generalizing the structure of universal enveloping algebras.
The document defines algebraic structures as collections of objects with operations that can be performed on them. It focuses on groups, rings, and fields. A group is defined as a set with an operation that satisfies four properties: closure, associativity, identity, and invertability. Examples of groups given include the integers, rational numbers, and real numbers under addition. It is noted that the integers also form an abelian group under addition. The document asks if the integers modulo 7 form a group and provides examples to verify group properties.
This document discusses L-fuzzy sub λ-groups. Key points:
- It defines L-fuzzy sub λ-groups and anti L-fuzzy sub λ-groups, which combine fuzzy set theory with lattice ordered group theory.
- Properties of L-fuzzy sub λ-groups are investigated, such as conditions under which a subset is a sub λ-group. The intersection of two L-fuzzy sub λ-groups is also an L-fuzzy sub λ-group.
- A relationship is established between an L-fuzzy sub λ-group and its complement, which must be an anti L-fuzzy sub λ-group.
- Level subsets of
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
The document appears to be a practice exam for a higher mathematics course covering topics like propositional logic, logical connectives, and lattices. It contains 20 multiple choice questions testing understanding of concepts like tautologies, partial orders, lattices, logical implications, and relationships between logical statements.
This document proves several theorems regarding positive integers and their properties related to greatest common divisors (GCDs). It first proves that the GCD of (abc, (a+b+c)(a^2 + b^2 + c^2)) is 1 if and only if the GCD of pairs of the integers and their sums is 1. It then extends this to prove a similar property for four integers a, b, c, d and their sums. Finally, it asks if a one-to-one mapping exists between very large odd prime numbers and the vertices of a graph such that the GCD of each prime and the sum of primes mapped to adjacent vertices is 1.
This document presents algorithms for generating all possible tree codes (representations of trees as sequences of integers) for a given number of edges. It establishes several arithmetic properties of tree generation codes, including that adding certain integer sequences to existing tree codes will always produce new valid tree codes. Theorems are provided to construct tree codes by appending, repeating, or transforming parts of existing tree codes in ways that preserve the tree structure. An algorithm is also described for checking if a given graceful graph code represents a tree using the Prüfer code technique.
This document discusses properties and generation of graceful graphs and trees through graceful codes. Some key points:
- Graceful codes represent graceful graphs through sequences of integers that satisfy certain properties. α-valuable codes and gracious codes are types of graceful codes.
- Properties of graceful/α-valuable codes are discussed, such as how codes can be combined or modified to generate new codes representing larger graphs.
- Theorems show how α-valuable codes of trees can be used to generate codes for larger trees through operations like duplication, insertion, or concatenation of codes. This provides an algorithmic way to systematically generate infinite families of graceful tree codes.
N. Chandramowliswaran was an inspired teacher who made difficult concepts transparent to students and was particularly strong in algebra. His colleague, Dr. K. Balasubramanian, believes Chandramowliswaran will shine as a teacher in any senior position based on the mental acumen and research abilities he demonstrated. Balasubramanian wishes Chandramowliswaran well for his promising future career.
This document summarizes a paper that proposes novel secret sharing and key distribution schemes based on number theory. It introduces generalized secret sharing schemes and key distribution schemes that are useful for multi-party systems. The paper then presents two main results: 1) A lemma showing that there exist integers that satisfy a particular congruence relation involving three distinct primes. 2) A theorem demonstrating how to generate secret shares of a secret S among three shareholders such that combining the shares recovers S. The scheme is based on the lemma and uses modular arithmetic with a modulus composed of three large primes.
This document summarizes key concepts in number theory presented in a lecture by Dr. N. Chandramowliswaran including:
- Properties of divisibility, greatest common divisors, prime numbers, and the fundamental theorem of arithmetic.
- Definitions and properties of the Mobius function, Euler's totient function, and congruences.
- Theorems regarding congruences including the Euler-Fermat theorem and the Chinese Remainder Theorem.
This document summarizes a paper that proposes novel secret sharing and key distribution schemes based on number theory. It introduces generalized secret sharing schemes and key distribution schemes that are useful for multi-party systems. The paper then presents two main results: 1) A lemma showing that there exist integers that satisfy a particular congruence relation involving three distinct primes. 2) A theorem demonstrating how to generate secret shares of a secret S among three shareholders such that combining the shares recovers S. The scheme is based on the lemma and uses modular arithmetic with a modulus composed of three large primes.
This document summarizes a paper that proposes novel secret sharing and key distribution schemes based on number theory. It introduces generalized secret sharing schemes and key distribution schemes that are useful for multi-party systems. The paper then presents two main results: 1) A lemma showing that there exist integers that satisfy a particular congruence relation involving three distinct primes. 2) A theorem demonstrating how to generate secret shares of a secret S among three shareholders such that combining the shares recovers S. The scheme is based on the lemma and uses modular arithmetic with a modulus composed of three large primes.
This document provides the schedule for a 7-day summer course on basic engineering mathematics, discrete mathematics, and graph theory held from July 6-13, 2009 at VIT University. The course schedule lists the daily topics to be covered from 9:45am-4:15pm, including lectures on graph theory, advanced calculus, formal languages and automata, probability and random processes, mathematical modeling, and programming fundamentals. Various professors from VIT University and other institutions are listed as lecturers for the different topics. The course is coordinated by Dr. N. Chandramowliswaran and overseen by the director Dr. K. Sathiyanarayanan.
This document describes an RSA encryption method using Pell's equation. It involves:
1) Selecting a secret odd prime integer R and finding the least positive integral solution (Y0, X0) to the Diophantine equation Y^2 - RX^2 = 1.
2) Selecting two large odd primes p and q and defining N = pq.
3) Defining the public key α using Y0, X0, R, and the Euler totient function φ(n).
The encryption of a message m involves computing E ≡ mS (mod n) where S is derived from α, and the decryption recovers m by computing E^d (mod n
The document discusses properties of tree generation codes and algorithms for generating tree codes. It defines graceful labeling and graceful codes for representing graceful graphs. Section 1 defines graceful codes and α-valuable codes. Section 2 presents arithmetic properties of tree generation codes, including theorems for generating tree codes by adding values to existing tree codes. Section 3 describes an algorithm to generate all tree codes for a given number of edges using Prüfer code techniques. The document also provides examples of tree codes.
This document proposes a secret key distribution technique using number theory. It presents a theorem that for any positive integer k ≥ 2, there exists k + k(k - 1)/2 share holders who can reconstruct a distributed secret. The technique selects secret positive integers and defines variables to distribute shares of the secret to the share holders, ensuring k share holders can reconstruct the secret. It concludes that cryptography has evolved over time into a mathematized discipline combining secret concealment with attack analysis.
1. Indian J. Pure Appl. Math., 43(2): 89-106, April 2012
c Indian National Science Academy
GROUP ALGEBRAS1
Inder Bir S. Passi
Dedicated to the memory of Indar Singh Luthar
Centre for Advanced Study in Mathematics, Panjab University,
Chandigarh 160014, India
and
Indian Institute of Science Education and Research,
Mohali (Punjab) 140306, India
e-mail: ibspassi@yahoo.co.in
(Received 3 February 2012; accepted 10 February 2012)
Given a group G and a commutative ring k with identity, one can define an
k-algebra k[G] called the group algebra of G over k. An element α ∈ k[G] is
said to be algebraic if f(α) = 0 for some non-zero polynomial f(X) ∈ k[X].
We will discuss some of the developments in the study of algebraic elements
in group algebras.
Key words : Group algebras, augmentation ideal, dimension sub-groups, al-
gebraic elements, partial augmentation, Jordan decomposition, idempotents,
Bass conjecture.
1
Prasanta Chandra Mahalanobis Medal 2011 Award Lecture delivered at Indian Institute of Sci-
ence Education and Research Mohali on 9 November 2011.
2. 90 INDER BIR S. PASSI
1. INTRODUCTION
Let k be a commutative ring with identity. An algebra A over k is a ring with
identity which is also a unital k-module with the following property:
(λa)b = a(λb) = λ(ab), for all λ ∈ k, and a, b ∈ A.
It is further required that the zero element of the module is the same element as the
zero element of the ring. For example, if k is a field, say, the field C of complex
numbers, then (i) the set k[X] of polynomials over k and (ii) the set Mn(k) of n×n
matrices over the field k are algebras over k with the usual addition, multiplication
and scalar multiplication.
Given a multiplicative group G and a commutative ring k with identity, the
set k[G] consisting of all the finite formal sums g∈G,α(g)∈k α(g)g with addition
defined coefficient-wise and multiplication induced by the multiplication in G to-
gether with distributivity is an algebra over k called the group algebra of the group
G over the commutative ring k. Thus, for α = g∈G α(g)g, β = h∈G β(h)h ∈
k[G],
α + β =
g∈G
(α(g) + β(g))g, αβ =
g∈G xy=g
α(x)β(y) g.
The element 1keG, where 1k is the identity element of the ring k and eG is the
identity element of G, is the identity element of the algebra k[G].
Observe that the map g → 1kg is a 1-1 group homomorphism from G into
the group of units of k[G], and the map λ → λeG is a 1-1 ring homomorphism
k → k[G]. We can thus identify both G and k with their respective images in
k[G] under the above maps. In particular, we then have 1k = eG = 1keG and this
element is the identity element of k[G] which we will denote by 1.
For a ring R with identity, let U(R) denote its group of units. If θ : G → U(A)
is a homomorphism from a group G into the group of units of a k-algebra A, then
it extends uniquely to an algebra homomorphism
θ : k[G] → A, α(g)g → α(g)θ(g).
3. GROUP ALGEBRAS 91
The trivial homomorphism G → U(k), g → 1, induces an algebra homomor-
phism, called the augmentation homomorphism,
: k[G] → k, α(g)g → α(g).
The ideal gk := ker of k[G] is called the augmentation ideal. In case k is the
ring Z of integers, we will drop the suffix k.
Example 1 : Let G = a be a finite cyclic group of order n, say. Then the
group algebra k[G] consists of the elements n−1
i=0 αiai, where αi ∈ k. Observe
that the ‘evaluation’ map
θ : k[X] → k[G], f(X) → f(a)
is an algebra homomorphism with kernel the ideal generated by Xn − 1. Thus we
have
k[G] k[X]/ Xn
− 1 .
Example 2 : Let G = a, b | a4 = 1, a2 = b2, b−1ab = a3 be the quaternion
group of order 8. Let HQ be the rational quaternion algebra. We have a homomor-
phism θ : Q[G] → HQ defined by a → i, b → j which is clearly an epimorphism.
Also, we have four epimorphisms θi : Q[G] → Q (i = 1, 2, 3, 4) induced by
a → ±1, b → ±1. Consequently we have an algebra homomorphism
ϕ : Q[G] → Q ⊕ Q ⊕ Q ⊕ Q ⊕ HQ.
It is easy to check that ϕ is an isomorphism.
Example 3 : Let D4 = a, b | a4 = 1 = b2, b−1ab = a3 be the dihedral group
of order 8. Then
Q[D4] Q ⊕ Q ⊕ Q ⊕ Q ⊕ M2(Q).
We have four homomorphisms Q[D4] → Q induced by a → ±1, b → ±1 and
a fifth homomorphism ρ : Q[D4] defined by
a →
0 1
−1 0
, b →
1 0
0 −1
.
4. 92 INDER BIR S. PASSI
My doctoral thesis [46] was written under the guidance of Professor David
Rees. Before taking up the main topic of this lecture, viz algebraic elements in
group algebras, I would like to mention one of the problems I investigated for my
thesis. If G is a group, then we have a descending central series {Dn(G)}n≥1 of
subgroups of G, called the dimension series, with
Dn(G) = G ∩ (1 + gn
), n ≥ 1,
and there arises the following:
Problem 1 : Determine the subgroups Dn(G), n ≥ 1.
The subgroup Dn(G) is called the nth dimension subgroup of G. For subsets
H, K of a group G, let [H, K] denote the subgroup generated by all elements of
the type h−1k−1hk with h ∈ H, k ∈ K. The lower central series of G is defined
inductively by setting
γ1(G) = G, γn+1(G) = [G, γn(G)], n ≥ 1.
The dimension series of any group G is closely related to its lower central
series. It is easily seen that Dn(G) ⊇ γn(G) for all groups G and all natural
numbers, n ≥ 1, and equality holds for n = 1, 2.
Theorem 1 (G. Higman, D. Rees; see [47]) — For evey group G, D3(G) =
γ3(G).
As a contribution to the above problem, I proved the following result [47].
Theorem 2 — If G is a finite p-group of odd order, then D4(G) = γ4(G).
The popular conjecture of the time was that the dimension and lower central
series of any group are identical. This conjecture was later refuted by Rips.
Theorem 3 [51] — There exists a group of order 238 for which D4(G) =
γ4(G).
5. GROUP ALGEBRAS 93
The study of dimension subgroups has continued to be one of my interests all
along. Interested readers may refer to the monographs [23], [40], [48].
I will now take up the discussion of algebraic elements. For an earlier survey
of this topic see [45].
Let A be an algebra over a a commutative ring k. An element α ∈ A is called
an algebraic element over k if there exists a non-constant polynomial f(X) ∈ k[X]
such that f(α) = 0. For example, if k is a field and A is a finite-dimensional alge-
bra over k, then clearly every element of A is an algebraic element. The algebraic
elements of most interest to algebraists are the nilpotent elements, idempotents and
torsion units; i.e., the elements which satisfy one of the following equations:
Xn
= 0, X2
− X = 0, Xn
− 1 = 0.
The term idempotent was introduced in 1870 by Benjamin Peirce in the context
of elements of an algebra that remain invariant when raised to a positive integer
power. The word idempotent literally means “(the quality of having) the same
power”, from idem + potence (same + power).
Our interest here is in the study of algebraic elements in group algebras. Recall
that, in a group G, an element x ∈ G is said to be conjugate to an element y ∈ G
if there exists z ∈ G such that
x = z−1
yz.
Conjugacy is an equivalence relation; the corresponding equivalence classes
are called the conjugacy classes of G. For each conjugacy class κ of G, the map
κ : k[G] −→ k defined by
κ(α) =
g∈κ
α(g)
is called the partial augmentation corresponding to the conjugacy class κ.
In the investigation of an algebraic element α = α(g)g ∈ k[G], the par-
tial augmentations κ(α) corresponding to the various conjugacy classes κ of G
provide a useful tool.
6. 94 INDER BIR S. PASSI
Let G be a finite group. A complex representation of G of degree n ≥ 1 is, by
definition, a homomorphism
ρ : G → GLn(C).
Let us recall some of the basic facts about complex representations of finite
groups. Two representations ρi : G → GLn(C) (i = 1, 2) are said to be equivalent
if there exists a matrix P ∈ GLn(C) such that
ρ1(g) = P−1
ρ2(g)P for all g ∈ G.
A representation ρ : G → GLn(C) is said to be reducible if there exists a
matrix P ∈ GLn(C) such that
P−1
ρ(g)P =
λ(g) 0
µ(g) ν(g)
for all g ∈ G,
where λ(G), µ(G), ν(G) are r × r, (n − r) × r, (n − r) × (n − r) matrices
respectively, and 1 ≤ r < n. Every finite group has exactly as many inequivalent
irreducible complex representations as the number of its distinct conjugacy classes.
The map χ : G → C given by
χ(g) = Trace(ρ(g)) (g ∈ G)
is called the character of the representation ρ. Note that if g, h ∈ G are conjugate
elements, then χ(g) = χ(h). Two representations of a finite group G are equivalent
if and only if the corresponding characters are equal.
Let G be a finite group, κ1, . . . , κk the conjugacy classes of G and χ(1), . . . , χ(k)
its full set of irreducible complex characters. Let χ
(i)
j denote the value that the
character χ(i) takes on the elements of the conjugacy class κj If κr, κs are two
conjugacy classes, then
k
i=1
χ(i)
r χ
(i)
s =
|G|
|κi|
δrs,
7. GROUP ALGEBRAS 95
where δij is the Kronecker delta and |S| denotes the cardinality of the set S. Let
χ : G → C be a character. Extend χ to the group algebra C[G] by linearity:
χ(α) =
g∈G
α(g)χ(g) (α =
g∈G
α(g)g).
Observe that we then have χ(α) = κ κ(α)χκ, where κ : C[G] → C is the
partial augmentation corresponding to the conjugacy class κ and χκ is the value
that the character χ takes on the elements of κ.
Let {si : si ∈ κi} be a system of representatives of the conjugacy classes
κi of a finite group G and let i : C[G] → C denote the partial augmentation
corresponding to the conjugacy class κi. Let α ∈ C[G] be an element satisfying a
polynomial f(X) ∈ C[X]. Suppose that λ is the maximum of the absolute values
of the roots of the polynomial f(X). If ρ is a representation with character χ, then
χ(α), being the trace of the matrix ρ(α), is a sum of χ(1) eigenvalues of ρ(α).
Consider the sum
S :=
1
|G| χ
χ(α)χ(α)
with χ running over the distinct irreducible characters of G. Because of the as-
sumption on λ, we have |χ(α)| ≤ χ(1)|λ|. Thus we conclude that
S ≤
1
|G| χ
χ(1)2
λ2
= λ2
.
On the other hand, note that
S =
1
|G|
χ , i , j
i(α) j(α)χ(si)χ(sj) =
i
| i(α)|2
/|κi|.
We thus have the following result.
Theorem 4 [24] — If G is a finite group, α ∈ C[G] is an element satisfying a
polynomial f(X) ∈ C[X] and λ is the maximum of the absolute values of the roots
of f(X), then
i
| i(α)|2
/|κi| ≤ λ2
.
8. 96 INDER BIR S. PASSI
Let A be a k-algebra. A map τ : A → k is called a trace if τ(ab) = τ(ba) for
all a, b ∈ A. Note that for every conjugacy class κ of a group G, the partial aug-
mentation κ : k[G] → k is a trace map. In particular, arising from the conjugacy
class of the identity element of G, the map 1 : k[G] → k, α → α(1) is a trace
map; this map is called the Kaplansky trace.
Theorem 5 [31, 38] — If e is an idempotent in the complex group algebra
C[G], then 1(e) is a totally real number, 0 ≤ 1(e) ≤ 1, and 1(e) = 0 if and only
if e = 0.
Kaplansky conjectured that if e ∈ C[G] is an idempotent, then 1(e) is a ra-
tional number. An affirmative answer to this conjecture was provided by A. E.
Zalesskii [58].
The above assertions are easily seen for finite groups. If G is a finite group and
e ∈ C[G] is such that e2 = e, then we have a decomposition C[G] = eC[G] ⊕
(1 − e)C[G]. Left multiplication by e defines a linear transformation τ : C[G] →
C[G] which is identity on the first component and zero on the second component.
Therefore the trace of the linear transformation τ equals the dimension of eC[G]
over C. On the other hand, if we compute the trace of τ using G as a basis, then
the trace works out to be 1(e)|G|. Consequently, we have
1(e) =
dimC(eC[G]).
|G|
Thus, for finite groups, we have a proof of Kaplansky’s theorem and the validity
of his conjecture.
An immediate consequence of Theorem 5 is that if α and β are in C[G] and
αβ = 1, then βα = 1. For, if αβ = 1, then βα is an idempotent. We thus have
1(βα) = 1(αβ) = 1. Hence, by Theorem 5, βα = 1.
An extension of the above result of Kaplansky, the subsequent bound given by
Weiss [57], and Theorem 4 is provided by the following result.
Theorem 6 [49] — If α is an element of C[G] satisfying the equation f(α) = 0
9. GROUP ALGEBRAS 97
for some non-zero polynomial f(X) ∈ C[X], and if λ denotes the maximum of the
absolute values of the complex roots of f(X), then
κ
| κ(α)|2
/|κ| ≤ λ2
,
where |κ| is the cardinality of the conjugacy class κ, κ(α)is the partial augmen-
tation of α with respect to κ.
For infinite groups we clearly cannot follow the approach of [24]. and so a
different method is needed. The above result has been extended to matrices over
C[G], by Luthar-Passi [33] (when G is finite) and to semisimple elements in (τ, ∗)-
algebras by Alexander [3] in a thesis written under the supervision of D. S. Pass-
man.
We next consider nilpotent elements in group algebras.
Theorem 7 ([5], Theorem 8.5; [49]) — If α ∈ C[G] is nilpotent, then κ(α) = 0
if either |κ| < ∞ or κ is a torsion conjugacy class.
Problem 2 — If α ∈ C[G] is nilpotent, must κ(α) be zero for every conjugacy
class κ of G?
Let us consider one source of nilpotent elements in group algebras. Let x, y ∈
G with x of finite order n, say. Then the element
α := (1 − x)y(1 + x + x2
+ · · · + xn−1
)
is clearly nilpotent: α2 = 0. Thus, if the group algebra k[G] has no non-zero
nilpotent elements, then α = 0, and therefore y = xyxi for some i, 0 ≤ i ≤ n−1.
It follows that if k[G] does not have non-zero nilpotent elements, then every finite
cyclic subgroup x , and therefore every subgroup of G if G is finite, must be
normal. Recall that a group all of whose subgroups are normal is called a Dedekind
group, and a non-abelian Dedekind group is called Hamiltonian. For the following
characterization of such groups, see [52].
Theorem 8 (Dedekind, Baer) — All the subgroups of a group G are normal
if and only if G is abelian or the direct product of a quaternion group of order 8,
10. 98 INDER BIR S. PASSI
an elementary abelian 2-group and an abelian group with all its elements of odd
order.
The following result gives a characterization of the rational group algebras
without nonzero nilpotent elements.
Theorem 9 [53] — The rational group algebra Q[G] of a finite group G has
no nonzero nilpotent element if and only if either the group G is abelian or it is a
Hamiltonian group of order 2nm with order of 2 mod m odd.
For algebraic number fields, in general, a complete answer has been given by
Arora [4]. If the group algebra k[G] of a finite group G over an algebraic number
field k has no nonzero nilpotent element, then, as seen above, every subgroup of
G must be normal and so G must either be abelian or a group of the type Q8 ⊕
E ⊕ A, where Q8 is the quaternion group of order 8, E is an elementary abelian
2-group, and A is an abelian group of odd order. For the characterization of finite
group algebras without nonzero nilpotent elements one thus needs to analyse the
structure of the group algebras of groups of the above kind over algebraic number
fields, and this is essentially a number-theoretic question. It is easily seen that in
this investigation one needs information about the stufe of fields, in particular, the
following result.
Theorem 10 [39] — Let Q(ζm) be a cyclotomic field, where ζm is a primitive
mth root of unity, m is odd and ≥ 3 . Then the equation −1 = x2 + y2 has a
solution x, y ∈ Q(ζm) if and only if the multiplicative order of 2 modulo m is
even.
Turning to idempotents, I would like to discuss a conjecture of H. Bass. To state
Bass’ conjecture, let us first recall the definition of the rank of a finitely generated
projective module.
Let R be a ring with identity and P a finitely generated projective right R-
module. The dual module P∗ = HomR(P, R) of P then carries a left R-module
structure and there exists an isomorphism
α : P ⊗R P∗
EndR(P)
11. GROUP ALGEBRAS 99
which is defined as follows. For x ∈ P and f ∈ P∗, α(x ⊗ f) : P → P is the
map y → xf(y), y ∈ P. Let [R, R] be the additive subgroup of R spanned by the
elements rs − sr (r, s ∈ R) and define
β : P ⊗R P∗
→ R/[R, R]
by setting β(x ⊗ f) = f(x) + [R, R].
The Hattori-Stallings rank (Hattori [25], Stallings [54]) of P, denoted rP is
defined to be β ◦α−1(1P ), where 1P is the identity endomorphism of P. Let T(G)
denote the set of conjugacy classes of the group G. Note that if R is the group
algebra k[G] of G over a commutative ring k with identity, then R/[R, R] is the
free k-module on the set T(G). Thus, we can write the rank of a finitely generated
projective k[G]-module P as
rP =
κ∈T(G)
rP (κ)κ.
Conjucture 1 ([5], Remark 8.2) — If P is a finitely generated projective C[G]-
module, then rP (κ) = 0 for the conjugacy classes of elements of infinite order.
This conjecture is still open.
For recent advancements in this direction see [6], [15], [16], [18], [19], [20],
[21], [37], [43], [55], [56].
A fundamental question in the theory of group algebras is the determination of
their structure and the computation of primitive central idempotents. Of particular
interest is the case of k[G] when k is a prime field, i.e., the field Q of rationals or
the finite field Fp of p elements for some prime p. My recent work with Gurmeet
K. Bakshi, Shalini Gupta and Ravi S. Kulkarni has been directed to this problem
([13], [7], [8], [12]).
Another problem concerning group rings that I have been interested in for some
time now is that of Jordan decomposition.
12. 100 INDER BIR S. PASSI
Let V be a finite-dimensional vector space over a perfect field k and α : V →
V a linear transformation. Then α can be written in one and only one way as
α = β + γ, β diagonalizable, γ nilpotent and βγ = γβ; furthermore, β, γ are
polynomials in α with coefficients from k. This decomposition of α is called its
Jordan decomposition. If G is a finite group, then every element α of the group
algebra k[G] induces a linear transformation from the k-vector space k[G] to itself
by left multiplication and hence has a Jordan decomposition as above, i.e., α can
be written in the form α = αs + αn, where αs and αn both lie in k[G] (and are
polynomials in α over k), αs is semisimple (i.e., satisfies a polynomial in k[X]
with no repeated roots), αn is nilpotent, and αsαn = αnαs. Thus, in particular,
if G is a finite group, then every element α of the rational group algebra Q[G]
has a unique Jordan decomposition. We say that the integral group ring Z[G] has
additive Jordan decomposition (AJD) if, for every element α ∈ Z[G], both the
semisimple and the nilpotent components αs, αn also lie in Z[G]. If α ∈ Q[G] is
an invertible element, then the semisimple component αs is also invertible and so
α = αsαu with αu (= 1 + α−1
s αn) unipotent and αsαu = αuαs. Furthermore,
such a decomposition is again unique. We say that the integral group ring Z[G]
has multiplicative Jordan decomposition (MJD) if both the semisimple component
αs and the unipotent component αu of every invertible element α ∈ Z[G] lie in
Z[G]. To characterize finite groups whose integral group rings have additive (resp.
multiplicative) Jordan decomposition is an interesting problem (see [29]).
With the complete answer for the characterization of finite groups whose inte-
gral group rings possess AJD having been available [28], the focus in recent years
has been on the multiplicative Jordan decomposition, a problem that is still unre-
solved.
Problem 3 — Characterize the finite groups G whose integral group rings Z[G]
possess multiplicative Jordan decomposition.
We summarize here the progress on this problem. Both the quaternion group
of order 8 and the dihedral group of order 8 have the MJD property [1].
13. GROUP ALGEBRAS 101
Out of the nine non-abelian groups of order 16, the integral group rings of
exactly five of them have the MJD property ([1], [44]).
There are exactly three non-abelian groups of order 32 whose integral group
ring have the MJD property but not the AJD property and, if G is a finite 2-group of
order at least 64, then Z[G] has the MJD property if and only if G is a Hamiltonian
group [30]. The case of finite 3-groups is almost settled by the following result.
Theorem 11 [34], [35] — The integral group rings of the two non-abelian
groups of order 27 have the multiplicative Jordan decomposition property; and
integral group rings of all other non-abelian groups of 3-power order do not have
this property except possibly the group of order 81 which is the central product of
a cyclic group of order 9 with the non-abelian group of order 27 of exponent 3.
Another recent progress towards the general solution is the following result.
Theorem 12 [35] — A finite 2, 3-group G with order divisible by 6 satisfies the
MJD property if and only if
(i) G = Sym3, the symmetric group of degree 3,
or
(ii) G = x, y | x3 = 1, y4 = 1, y−1xy = x−1 , a group of order 12,
or
(iii) G = Q8 × C3, the direct product of the quaternion group of order 8 with
the cyclic group of order 3.
For another perspective about the notion of Jordan decomposition see [50].
Finally, let us consider torsion units. Let G be a finite abelian group and let
α = g∈G α(g)g ∈ Z[G] be such that g∈G α(g) = 1 and αn = 1 for some
n ≥ 1, then, by Theorem 4, g∈G α(g)2 ≤ 1. Consequently, α(g) is non-zero
for exactly one element g ∈ G, and for that element α(g) = 1. We thus have the
following well-known result.
14. 102 INDER BIR S. PASSI
Theorem 13 [27] — If G is a finite abelian group, then the torsion units in its
integral group ring Z[G] are exactly the elements ±g (g ∈ G).
For arbitrary groups, the following conjecture is still unresolved.
Conjecture 2 — If G is a finite group and α ∈ Z[G] is a torsion unit of augmen-
tation 1, then there exists an invertible element β ∈ Q[G] and an element g ∈ G
such that β−1αβ = g.
Luthar-Passi [32] verified Zassenhaus conjecture for the alternating group A5.
Their method has since been used successfully by various authors to confirm this
conjecture in several other cases (see e.g., [9], [10], [11], [26], [36]).
For an extension of the above result to torsion units in matrix group rings see
[14], [33], [41], [42].
ACKNOWLEDGMENT
I am indebted to the Indian National Science Academy for the award of the Prasanta
Chandra Mahalabnobis Medal 2011. I would like to express my most grateful
thanks to the Academy for the honor bestowed on me.
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