Finite Abelian Group Research Papers (original) (raw)

The fixing number of a graph G is the smallest cardinality of a set of vertices S such that only the trivial automorphism of G fixes every vertex in S. The fixing set of a group is the set of all fixing numbers of finite graphs w ith automorphism group . Several authors have studied the distinguishing nu mber of a graph, the smallest number of labels needed to label G so that the automorphism group of the labeled graph is trivial. The fixing number can be thought of as a variation of the distinguishing number in which every label may be used only once, and not every vertex need be labeled. We characterize the fixing sets of finite abelian groups, and investigate the fixing sets of symmetric groups.

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- Pure Mathematics, Symmetric group, Automorphism P-Group, Finite Abelian Group

We use the theory of cubic structures to give a fixed point Riemann-Roch formula for the equivariant Euler characteristics of coherent sheaves on projective flat schemes over Z with a tame action of a finite abelian group. This formula supports a conjecture concerning the extent to which such equivariant Euler characteristics may be determined from the restriction of the sheaf to an infinitesimal neighborhood of the fixed point locus. Our results are applied to study the module structure of modular forms having Fourier coefficients in a ring of algebraic integers, as well as the action of diamond Hecke operators on the Mordell-Weil groups and Tate-Shafarevich groups of Jacobians of modular curves. Contents §1. Introduction §2. Cubic structures and categories §3. The main calculation and result §4. Galois structure of modular forms §5. An equivariant Birch and Swinnerton-Dyer relation 1 Corollary 1.5. Suppose θ 2 [P χ ] = 0 in Cl(Z[ζ r , 1 2 ]). Then at least one of s χ ([J ′ H (Q)]) or s χ ([X(J H )]) is non-trivial if the Birch Swinnerton-Dyer conjecture of §5 holds. Suppose in addition that C(p) = J 1 (p)(Q) tor , as conjectured in [CES]. Then either the χ-eigenspace of

- by G. Pappas
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- Number Theory, Algebraic Geometry, Pure Mathematics, Fixed Point Theory

Let G be a finite abelian group and A be a subset of G. We say that A is complete if every element of G can be represented as a sum of different elements of A. In this paper, we study the following question What is the structure of a... more

- by Hung Vu
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- Number Theory, Mathematical Sciences, Finite Abelian Group

The second author introduced notions of weak permutablity and commutativity between groups and proved the finiteness of a group generated by two weakly permutable finite subgroups. Two groups H, K weakly commute provided there exists a bijection f : H → K which fixes the identity and such that h commutes with its image h f for all h ∈ H . The present paper gives support to conjectures about the nilpotency of groups generated by two weakly commuting finite abelian groups H, K .

- by Said Sidki
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- Group Theory, Finite Group Theory, Pure Mathematics, Finite Abelian Group

Generalized Clifford algebras (GCAs) and their physical applications were extensively studied for about a decade from 1967 by Alladi Ramakrishnan and his collaborators under the name of L-matrix theory. Some aspects of GCAs and their physical applications are outlined here. The topics dealt with include: GCAs and projective representations of finite abelian groups, Alladi Ramakrishnan's sigma operation approach to the representation theory of Clifford algebra and GCAs, Dirac's positive energy relativistic wave equation, Weyl-Schwinger unitary basis for matrix algebra and Alladi Ramakrishnan's matrix decomposition theorem, finite-dimensional Wigner function, finite-dimensional canonical transformations, magnetic Bloch functions, finite-dimensional quantum mechanics, and the relation between GCAs and quantum groups.

The partial group algebra of a group G over a field K, denoted by Kpar(G), is the algebra whose representations correspond to the partial representations of G over K-vector spaces. In this paper we study the structure of the partial group algebra Kpar(G), where G is a finite group. In particular, given two finite abelian groups G1 and G2, we prove that if the characteristic of K is zero, then Kpar(G1) is isomorphic to Kpar(G2) if and only if G1 is isomorphic to G2.

- by Michael Dokuchaev
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- Mathematics, Algebra, Group Theory, Finite Group Theory

A sequence of elements of a finite group G is called a zero-sum sequence if it sums to the identity of G. The study of zero-sum sequences has a long history with many important applications in number theory and group theory. In 1989 Kleitman and Lemke, and independently Chung, proved a strengthening of a number theoretic conjecture of Erdos and Lemke. Kleitman and Lemke then made more general conjectures for finite groups, strengthening the requirements of zero-sum sequences. In this paper we prove their conjecture in the case of abelian groups. Namely, we use graph pebbling to prove that for every sequence (g_k)_{k=1}^{|G|} of |G| elements of a finite abelian group G there is a nonempty subsequence (g_k)_{k in K} such that sum_{k in K}g_k=0_G and sum_{k in K}1/|g_k|\le 1, where |g| is the order of the element g in G.

- by Shawn Elledge and +1
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- Number Theory, Group Theory, Finite Group Theory, Finite Abelian Group

Quantum states are very delicate, so it is likely some sort of quantum error correction will be necessary to build reliable quantum computers. The theory of quantum error-correcting codes has some close ties to and some striking differences from the theory of classical error-correcting codes. Many quantum codes can be described in terms of the stabilizer of the codewords. The stabilizer is a finite Abelian group, and allows a straightforward characterization of the error-correcting properties of the code. The stabilizer formalism for quantum codes also illustrates the relationships to classical coding theory, particularly classical codes over GF(4), the finite field with four elements.

- by Lorenza Viola
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- Coding Theory, Quantum Physics, Multidisciplinary, Biological Sciences

In this semitutorial paper we discuss a general message passing algorithm, which we call the generalized distributive law (GDL). The GDL is a synthesis of the work of many authors in the information theory, digital communications, signal processing, statistics, and artificial intelligence communities. It includes as special cases the Baum-Welch algorithm, the fast Fourier transform (FFT) on any finite Abelian group, the Gallager-Tanner-Wiberg decoding algorithm, Viterbi's algorithm, the BCJR algorithm, Pearl's "belief propagation" algorithm, the Shafer-Shenoy probability propagation algorithm, and the turbo decoding algorithm. Although this algorithm is guaranteed to give exact answers only in certain cases (the "junction tree" condition), unfortunately not including the cases of GTW with cycles or turbo decoding, there is much experimental evidence, and a few theorems, suggesting that it often works approximately even when it is not supposed to.

- by Aji Aji
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- Information Theory, Signal Processing, Digital Communication, Artificial Intelligent
- by Roger Alperin
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- Pure Mathematics, Finite Abelian Group

The topic of this paper is (multi-window) Gabor frames for signals over finite Abelian groups, generated by an arbitrary lattice within the finite timefrequency plane. Our generic approach covers both multi-dimensional signals as well as non-separable lattices. The main results reduce to well-known fundamental facts about Gabor expansions of finite signals for the case of product lattices, as they have been given by Qiu, Wexler-Raz or Tolimieri-Orr, Bastiaans and Van-Leest, among others. In our presentation a central role is given to spreading function of linear operators between finite-dimensional Hilbert spaces. Another relevant tool is a symplectic version of Poisson's summation formula over the finite time-frequency plane. It provides the Fundamental Identity of Gabor Analysis. In addition we highlight projective representations of the time-frequency plane and its subgroups and explain the natural connection to twisted group algebras. In the finite-dimensional setting these twisted group algebras are just matrix algebras and their structure provides the algebraic framework for the study of the deeper properties of finite-dimensional Gabor frames. c → i∈I c i g i .

- by Franz Luef
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- Applied Mathematics, Group Theory, Pure Mathematics, Structuration Theory

Descriptive complexity theory is a branch of complexity theory that views the hardness of a problem in terms of the complexity of expressing it in some logical formalism; among the resources considered are the number of object variables, quantifier depth, type, and alternation, sentences length (finite/infinite), etc. In this field we have studied two problems: (i) expressibility in ∃SO and (ii) the descriptive complexity of finite abelian groups. Inspired by Fagin’s result that NP = ∃SO, we have developed a partial framework to investigate expressibility inside ∃SO so as to have a finer look into NP. The framework uses combinatorics derived from second-order Ehrenfeucht-Fraısse games and the notion of game types. Among the results obtained is that for any k, divisibility by k is not expressible by an ∃SO sentence where (1) each second-order variable has arity at most 2, (2) the first-order part has at most 2 first-order variables, and (3) the first-order part has quantifier depth at most 3. In the second project we have investigated the descriptive complexity of finite abelian groups. Using Ehrenfeucht-Fraısse games we find upper and lower bounds on quantifier depth, quantifier alternations, and number of variables of a first-order sentence that distinguishes two finite abelian groups. Our main results are the following. Let G1 and G2 be a pair of non-isomorphic finite abelian groups, and let m be a number that divides one of the two groups’ orders. Then the following hold: (1) there exists a first-order sentence φ that distinguishes G1 and G2 such that φ is existential, has quantifier depth O(log m), and has at most 5 variables and (2) if φ is a sentence that distinguishes G1 and G2 then φ must have quantifier depth Ω(log m). In infinitary model theory we have studied abstract elementary classes. We have defined Galois types over arbitrary subsets of the monster (large enough homogeneous model), have defined a simple notion of splitting, and have proved some properties of this notion such as invariance under isomorphism, monotonicity, reflexivity, existence of non-splitting extensions.

- by Walid Gomaa
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- Model Theory, Computer Science, Computational Complexity, Complexity Theory

- by Muhammad Aji
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- Mathematics, Computer Science, Artificial Intelligence, Information Theory

... I'd like also to thank Mohamed Shawky, Mohamed Abdallah, Ahmed Sadek, Karim Sadik, Tarek Ghanem, Yasser Jaradat, Mohamed Farouk, Amr ElSherif and Mohamed Fahmi. ii Page 10. Table of Contents 1 Introduction 1 1.1 Preliminary .... more

- by Walid Gomaa
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- Model Theory, Computer Science, Computational Complexity, Complexity Theory

Let Π be a finite projective plane admitting a large abelian collineation group. It is well known that this situation may be studied by algebraic means (via ar epresentation by suitable types of difference sets), namely using group rings and algebraic number theory and leading to rather strong nonexistence results. What is less well-known is the fact that the abelian

- by Dina Ghinelli
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- Number Theory, Group Ring Theory, Finite Abelian Group
- by M. Zaicev and +1
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- Algebra, Pure Mathematics, Finite Abelian Group, Matrix Algebra

- by Glenn Hurlbert
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- Mathematics, Number Theory, Group Theory, Finite Group Theory

The power graph of a group is the graph whose vertex set is the group, two elements being adjacent if one is a power of the other. We observe that nonisomorphic finite groups may have isomorphic power graphs, but that finite abelian groups with isomorphic power graphs must be isomorphic. We conjecture that two finite groups with isomorphic power graphs have the same number of elements of each order. We also show that the only finite group whose automorphism group is the same as that of its power graph is the Klein group of order 4.

- by Peter Cameron
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- Applied Mathematics, Combinatorics, Group Theory, Finite Group Theory

For G a finite abelian group, we study the properties of general equivalence relations on Gn = G n ⋊Sn, the wreath product of G with the symmetric group Sn, also known as the G-coloured symmetric group. We show that under certain conditions, some equivalence relations give rise to subalgebras of kGn as well as graded connected Hopf subalgebras of n≥o kGn. In particular we construct a G-coloured peak subalgebra of the Mantaci-Reutenauer algebra (or G-coloured descent algebra). We show that the direct sum of the G-coloured peak algebras is a Hopf algebra. We also have similar results for a G-colouring of the Loday-Ronco Hopf algebras of planar binary trees. For many of the equivalence relations under study, we obtain a functor from the category of finite abelian groups to the category of graded connected Hopf algebras. We end our investigation by describing a Hopf endomorphism of the G-coloured descent Hopf algebra whose image is the G-coloured peak Hopf algebra. We outline a theory of combinatorial G-coloured Hopf algebra for which the G-coloured quasi-symmetric Hopf algebra and the graded dual to the G-coloured peak Hopf algebra are central objects.

- by Christophe Hohlweg
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- Algebraic Combinatorics, Pure Mathematics, Symmetric group, Binary Tree

We consider the question: When do two finite abelian groups have isomorphic lattices of characteristic subgroups? An explicit description of the characteristic subgroups of such groups enables us to give a complete answer to this question in the case where at least one of the groups has odd order. An "exceptional" isomorphism, which occurs between the lattice of characteristic subgroups of Z p × Z p 2 × Z p 4 and Z p 2 × Z p 5 , for any prime p, is noteworthy.

- by Brent Kerby
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- Group Theory, Pure Mathematics, Finite Abelian Group

A geometric object is called reflexible or chiral as it is or is not isomorphic to its mirror image. We introduce the chirality group and the chirality index of an arbitrary regular hypermap, as algebraic and numerical measures of its lack of mirror symmetry. We establish some general properties of chirality groups, and show that certain classes of groups may or may not arise in this way.

- by Antonio Breda D'Azevedo
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- Group Theory, Algebraic Combinatorics, Pure Mathematics, Indexation

Two problems are considered. First, the conjecture that all odd abelian groups except H,, Es, H,, and Z, + Z, admit strong starters, is reduced to finding strong starters in five types of groups; the cyclic groups of order 3p, 9p, 3' for k > 6, 5 .3" for k > 4, and ZL, + hs, where p is any odd prime greater than 111. It is shown that all abelian groups G of odd order greater than 5 such that three does not divide the order of G admits a strong starter. As well, strong starters are given in some small non-cyclic groups which were previous not known to admit starters. Second, a multiplication theorem for sets of pairwise orthogonal starters is given. An exhaustive computer search for orthogonal starters in odd groups smaller than 19 is carried out. The results require the construction of special permutations of some groups.

- by Joseph Horton
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- Mathematics, Applied Mathematics, Computer Science, Pure Mathematics

In this paper we describe all gradings by a finite abelian group G on the following Lie algebras over an algebraically closed field F of characteristic p = 2: sl n (F) (n not divisible by p), so n (F) (n ≥ 5, n = 8) and sp n (F) (n ≥ 6, n... more

- by Yuri Bahturin
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- Mathematics, Lie Algebra, Pure Mathematics, Lie Group

In this paper we obtain an explicit formula for the higher Schurmultiplicator of an arbitrary finite abelian group with respect to the variety of nilpotent groups of class at most c ≥ 1 .

- by Mohammad Reza R. Moghaddam
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- Finite Abelian Group

In this paper we describe all group gradings by a finite Abelian group G of several types of simple Jordan and Lie algebras over an algebraically closed field F of characteristic zero.  2004 Published by Elsevier Inc. (Y.A. Bahturin),... more

- by M. Zaicev and +2
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- Algebra, Lie Algebra, Pure Mathematics, Finite Abelian Group

In this paper we describe all group gradings by a finite abelian group G of any Lie algebra L of the type "A" over algebraically closed field F of characteristic zero.

- by M. Zaicev and +1
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- Lie Algebra, Finite Abelian Group
- by Sándor Szabó
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- Applied Mathematics, Algebra, Pure Mathematics, Discrete Mathematics

We describe a general framework in which we can precisely compare the structures of quantum-like theories which may initially be formulated in quite different mathematical terms. We then use this framework to compare two theories: quantum mechanics restricted to qubit stabiliser states and operations, and Spekkens's toy theory. We discover that viewed within our framework these theories are very similar, but differ in one key aspect–a four element group we term the phase group which emerges naturally within our framework. In the case ...

- by Bob Coecke
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- Cognitive Science, Quantum Physics, Category Theory, Quantum Mechanics

- by Muhammad Aji
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- Mathematics, Computer Science, Artificial Intelligence, Information Theory

The critical group of a graph is a finite abelian group whose order is the number of spanning trees of the graph. There is a simple relationship between the number of spanning trees for a graph and for its line graph whenever the latter is regular. This suggests a simple relationship between their critical groups. We present such a relationship in the form of an exact sequence.

- by Victor Reiner
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- Super Line Graph, Spanning Tree, Finite Abelian Group, Regular graph

In the paper, we study special configurations of lines and points in the complex projective plane, so called k-nets. We describe the role of these configurations in studies of cohomology on arrangement complements. Our most general result... more

- by Sergey Yuzvinsky
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- Algebraic Geometry, Pure Mathematics, Symplectic geometry, Finite Abelian Group

The paper describes a simple procedure for computing the second cohomology H 2 (G A) of a nitely presented nite group G with coecients in a nitely presented nite abelian group A. An accompanying text-le contains a Magma implementation of... more

- by Irina Kholodna
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- Finite Group Theory, Pure Mathematics, Finite Abelian Group

A function on a (generally infinite) graph Γ with values in a field K of characteristic 2 will be called harmonic if its value at every vertex of Γ is the sum of its values over all adjacent vertices. We consider binary pluri-periodic harmonic functions f : Z s → F 2 = GF(2) on integer lattices, and address the problem of describing the set of possible multi-periodsn = (n 1 ,. .. , n s) ∈ N s of such functions. This problem arises in the theory of cellular automata [O.

- by Mikhail Zaidenberg
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- Applied Mathematics, Automata, Cellular Automata, Complex System

We propose public-key primitives and schemes based on IQ-related com- putational problems for standardisation and deployment. IQ refers to class groups of imaginary quadratic orders, and IQ-cryptography to schemes using these groups. IQ-cryptography oers another secure and ecient alternative to other forms of pub- lic key cryptography. Class groups of imaginary quadratic number fields are finite Abelian groups for which

- by Safuat Hamdy
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- Standardisation, Finite Abelian Group, Signature Scheme, Discrete Logarithm

This paper is devoted to the correction of an error in the paper [5] in which the classification of involution gradings on matrix algebras was derived from the fact that in the decomposition of a graded matrix algebra as the tensor product of an elementary and a fine component, these components remain invariant under the involution. 2 Some notation and simple facts Let F be an arbitrary field, A a not necessarily associative algebra over an F and G a group. We say that A is a G-graded algebra, if there is a vector space

- by Yuri Bahturin
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- Algebra, Pure Mathematics, Finite Abelian Group, Matrix Algebra

In this paper we discuss a source of finite abelian groups suitable for cryptosystems based on the presumed intractability of the discrete logarithm problem for these groups. They are the jacobians of hyperelliptic curves defined over finite fields. Special attention is given to curves defined over the field of two elements. Explicit formulas and examples are given, and the problem of finding groups of almost prime order is discussed.

An algorithm for calculating a set of generators of representative 2-cocycles on semidirect product of finite abelian groups is constructed, in light of the theory over cocyclic matrices developed by Horadam and de Launey in . The method involves some homological perturbation techniques [3, 1], in the homological correspondent to the work which Grabmeier and Lambe described in from the viewpoint of cohomology. Examples of explicit computations over all dihedral groups D4t are given, with aid of Mathematica.

- by Victor Jelves
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- Algebra, Homological Algebra, Cohomology, Finite Abelian Group

We present new algorithms for computing orders of elements, discrete logarithms, and structures of finite abelian groups. We estimate the computational complexity and storage requirements, and we explicitly determine the O O -constants and Ω \Omega -constants. We implemented the algorithms for class groups of imaginary quadratic orders and present a selection of our experimental results. Our algorithms are based on a modification of Shanks’ baby-step giant-step strategy, and have the advantage that their computational complexity and storage requirements are relative to the actual order, discrete logarithm, or size of the group, rather than relative to an upper bound on the group order.

- by Michael John Jacobson, Jr.
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- Mathematics, Number Theory, Applied Mathematics, Computer Science
- by Andrew Goodall
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- Mathematics, Computer Science, Graph Theory, Pure Mathematics

- by hung vu
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- Mathematics, Number Theory, Computer Science, Mathematical Sciences

We obtain efficient sampling methods for recovering or compressing functions over finite Abelian groups with few Fourier coefficients, i.e., functions that are (approximable by) linear combinations of few, possibly unknown Fourier basis functions or characters. Furthermore, our emphasis is on efficiently and deterministically finding small, uniform sample sets, which can be used for sampling all functions in natural approximation classes of Boolean functions. Due to this requirement, even the simplest versions of this problem (say, when the set of approximating characters is known) require somewhat different techniques from the character theory of finite Abelian groups that are commonly used in other discrete Fourier transform applications. We briefly discuss applications of our efficient, uniform sampling methods in computational learning theory, efficient generation of pseudorandom strings, and testing linearity; we also state highly related open problems that are not only applicable in these contexts, but are also of independent mathematical interest.

- by Meera Sitharam
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- Mathematics, Computer Science, Data Compression, Sampling methods

- by Martin Škoviera
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- Group Theory, Algebraic Combinatorics, Pure Mathematics, Indexation