Symmetric group

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.

In this paper we investigate the minimum number of maximal subgroups H_i for i=1 ...k of the symmetric group S_n (or the alternating group A_n) such that each element in the group S_n (respectively A_n) lies in some conjugate of one of the H_i. We prove that this number lies between a.phi(n) and bn for certain constants a, b, where phi(n) is the Euler phi-function, and we show that the number depends on the arithmetical complexity of n. Moreover in the case where n is divisible by at most two primes, we obtain an upper bound of 2+phi(n)/2, and we determine the exact value for S_n when n is odd and for A_n when n is even.

The most common concern of any communication system is the data quality. There exist different components that can impact the quality of data during its conveying over the channel as noise, fading, etc. Forward error correcting codes (FEC) play a major role for overcoming this noise as it adds a control bits to the original data for error detection and correction. This paper aims at analyzing convolutional codes with different rates and evaluating its performance. Binary phase shift modulation (BPSK) scheme and binary symmetric channel (BSC) model are used. First a convolution encoder is presented and then additive white Gaussian noise (AWGN) is added. The paper uses maximum likelihood mechanism (Vetribi Algorithm) for decoding process. Simulations are carried out using MATLAB with Simulink tools. Bit error rate (BET) is used as testing parameter and results of system behavior for both coded and encoded are compared.

Abstract: Encryption is the most effective way to achieve data security. It is the process in which plain text converts into a cipher text and allows only authorized people to access the sender information. In this research paper I would like to explain different types of encryption techniques such as hashing function and symmetric and asymmetric methods with advantages and disadvantages including necessary practical examples.
Keywords: Encryption, Cryptography, Hashing, Asymmetric, Symmetric, Cipher text
Title: Encryption
Author: Sruthi Medasani, Professor Tarik El Taeib,
International Journal of Computer Science and Information Technology Research
ISSN 2348-1196 (print), ISSN 2348-120X (online)
Research Publish Journals

A group key agreement (GKA) protocol allows a set of users to establish a common secret via open networks. Observing that a major goal of GKAs for most applications is to establish a confidential channel among group members, we revisit the group key agreement definition and distinguish the conventional (symmetric) group key agreement from asymmetric group key agreement (ASGKA) protocols. Instead of a common secret key, only a shared encryption key is negotiated in an ASGKA protocol. This encryption key is accessible to attackers and corresponds to different decryption keys, each of which is only computable by one group member. We propose a generic construction of one-round ASGKAs based on a new primitive referred to as aggregatable signature-based broadcast (ASBB), in which the public key can be simultaneously used to verify signatures and encrypt messages while any signature can be used to decrypt ciphertexts under this public key. Using bilinear pairings, we realize an efficient ASBB scheme equipped with useful properties. Following the generic construction, we instantiate a one-round ASGKA protocol tightly reduced to the decision Bilinear Diffie-Hellman Exponentiation (BDHE) assumption in the standard model.

We say that a group $G$ has Bergman's property (the property of universality of finite width) if for every generating set $X$ of $G$ with $X=X^{-1}$ we have that $G=X^k$ for some natural number $k.$ The property is named after George Bergman who have proved recently that the infinite symmetric groups are groups of universally finite width. The first example of an infinite group with Bergman's property is due to Shelah (1980s). Lately some other examples have been found: the automorphism groups of doubly transitive chains (Droste-Holland), the automorphism group of reals as a Borel space (Droste-G\"obel) and the infinite-dimensional general linear groups over division rings. In the present paper we prove that the automorphism group $Aut(N)$ of any infinitely generated free nilpotent group $N$ has Bergman's property, is generated by involutions, perfect and has confinality greater than $rank(N).$ Also, we obtain a partial answer to a question posed by Bergman establis...

Polynomials with values in an irreducible module of the symmetric group can be given the structure of a module for the rational Cherednik algebra, called a standard module. This algebra has one free parameter and is generated by differential-difference ("Dunkl") operators, multiplication by coordinate functions and the group algebra. By specializing Griffeth's (arXiv:0707.0251) results for the G(r,p,n) setting, one obtains norm formulae for symmetric and antisymmetric polynomials in the standard module. Such polynomials of minimum degree have norms which involve hook-lengths and generalize the norm of the alternating polynomial. Comment: 22 pages, added remark about the Gordon-Stafford Theorem, corrected some typos

Summary. A new class of prior distributions for metric-based models in the analysis of fully and partially ranked data is developed. This class is attractive because it provides a meaningful way to encapsulate prior information about the parameters of the model. Three examples illustrate the ideas developed in the paper.

Many people experience an inner delight when beholding regularly shaped crystals, a quiet reverence for their wonderfully regular forms. And thus the question soon arises regarding the nature of crystals, the lawfulness underlying their shapes and properties. In textbooks, popular writings and in museum exhibitions, mineralogists and crystallographers mostly emphasize the lattice structure of crystalline matter, which is said to underlie the vast variety of crystal phenomena. Yet a certain disappointment may be felt as a result of being unable to bring one’s inner experience into any sort of deeper relationship with these lattice structures. This essay will draw attention to the fact that the assumption of a lattice structure only expresses one aspect of a crystal’s nature, and consequently that it can and must be embedded in a more comprehensive relationship.
Phenomenological and experimental investigations of crystals reveal that the majority of their physical and geometrical properties are dependent on entirely distinct spatial directions, immanent in the individual crystals themselves. For example, the level of hardness and the formation of planar cleavages are not the same in all directions; they have maxima and minima in different directions, thus laying the foundation for an initial (inner) orientation of the crystal. Similar properties are exhibited in the thermal expansion, the electrical conductivity, the piezoelectricity, the elastic vibrations, as well as in the refraction of light and polarization phenomena of non-cubic crystallizing minerals.
Here the so-called Neumann principle holds: the symmetries of the physical properties of a crystal at least contain the geometrical symmetries of the corresponding crystal polyhedron. This means that a geometrical examination of the shapes and symmetries of a crystal within the context of geometric crystal morphology furnishes the higher order symmetries that underlie all the symmetries of the specific physical properties. Hence symmetries (especially axes and planes of symmetry) are naturally occurring properties, which may indeed be discovered and ascertained in a specific, finite crystal polyhedron, yet point beyond the boundaries of the individual crystal. For on the one hand the same symmetries may be attributed to different crystals, while on the other the axes and planes of symmetry point beyond the finite crystal body to the entire surrounding space. This opens up the conceptual possibility of no longer simply comprehending crystals in a local-additive manner, as structures solely arising from the reciprocal effects of elementary particles. Therefore crystals may also be understood as distinctly unified forms, having their origin in non-localized formative principles, which, considered geometrically, correspond to configurations spanning the whole of space.
Among other things the present investigation is a case study of a central problem – how to undertake an anthroposophical/spiritual-scientific extension of traditional natural-scientific knowledge. Using crystallography, it will be shown how out of a “sound instinct for knowledge” (Steiner (1918), p. 107), crystal physicists have concentrated on the inner, objectively conditioned necessity of one of the fundamental assumptions of scientific crystallography. This at once paves the way for an extension into the conceptual possibility of spiritual forces  yet without coming into conflict with any of the fundamental laws of physics. This fundamental assumption concerns the infinite nature of a crystal lattice (see section 6). It also opens the door to so-called reciprocal space, which plays such a vital role in solid-state physics (section 7).
To begin with, the present study provides a summary of the most important morphological viewpoints (section 2) and the classic crystallographic laws (section 3), as well as outlining the systematic construction of the theory of crystal forms (section 4). Section 5 on morphology and structure theory examines the two complementary interpretations of the fundamental law of crystallography, the lattice structure hypothesis, and the crystal morphological hypothesis. The relationship between the two will be discussed in greater detail in section 6. This latter hypothesis will require some projective geometry, which will be considered in section 5. The article concludes with a brief look at a number of physical considerations (section 7), as well as the structure and morphogenesis of crystals and their synthesis (section 8).

The set of nonnegative integers $W$ is mapped bijectively to the finitary permutation group $FS(N)$ on the set of natural numbers, using the factoradic expansion of an integer and, broadly interpreting Knuth, any exhaustive permutation-generation algorithm.  One such mapping using minimal composition of transpositions is presented, and this mapping is used to define a nomenclature scheme. This scheme is used for representation and application of permutations within the structure of common computer-scientific uses, while determining other computational uses. The mapping is also extended to determine a nomenclature scheme for the entire permutation group $S_N$ on the set of natural numbers.

Our results include many new constructions based on strong twisted union and wreath product, with an investigation of retracts and the multipermutation level and the solvable length of the groups defined by the solutions and new results about decompositions and factorisations of the groups defined by invariant subsets of the solution.

The commuting graph ${\cal C}(G,X)$, where $G$ is a group and $X$ a subset of $G$, has $X$ as its vertex set with two distinct elements of $X$ joined by an edge when they commute in $G$. Here the diameter and disc structure of ${\cal C}(G,X)$ is investigated when $G$ is the symmetric group and $X$ a conjugacy class of $G$.

The fusion procedure provides a way to construct new solutions to the Yang-Baxter equation. In the case of the symmetric group the fusion procedure has been used to construct diagonal matrix elements using a decomposition of the Young diagram into its rows or columns. We present a new construction which decomposes the diagram into hooks, the great advantage of this is that it minimises the number of auxiliary parameters needed in the procedure. We go on to use the hook fusion procedure to find diagonal matrix elements computationally and calculate supporting evidence to a previous conjecture. We are motivated by the construction of certain elements that allow us to generate representations of the symmetric group and single out particular irreducible components. In this way we may construct higher representations of the symmetric group from elementary ones. We go some way to generalising the hook fusion procedure by considering other decompositions of Young diagrams, specifically int...

Using these nonstandard objects as a guide, we follow the approach of Adsul, Sohoni, and Subrahmanyam to construct, in the case dim(V) = dim(W) =2, a representation \check{X}_\nu of the nonstandard quantum group that specializes to Res_{GL(V) \times GL(W)} X_\nu at q=1. We then define a global crystal basis +HNSTC(\nu) of \check{X}_\nu that solves the two-row Kronecker problem: the number of highest weight elements of +HNSTC(\nu) of weight (\lambda,\mu) is the Kronecker coefficient g_{\lambda \mu \nu}. We go on to develop the beginnings of a graphical calculus for this basis, along the lines of the U_q(\sl_2) graphical calculus, and use this to organize the crystal components of +HNSTC(\nu) into eight families. This yields a fairly simple, explicit and positive formula for two-row Kronecker coefficients, generalizing a formula of Brown, van Willigenburg, and Zabrocki. As a byproduct of the approach, we also obtain a rule for the decomposition of Res_{GL_2 \times GL_2 \rtimes §_2} X_\nu into irreducibles.

(Abridged abstract) For a finite real reflection group W and a W-orbit O of flats in its reflection arrangement---or equivalently a conjugacy class of its parabolic subgroups---we introduce a statistic on elements of W. We then study the operator of right-multiplication within the group algebra of W by the element whose coefficients are given by this statistic. We reinterpret the operators geometrically in terms of the arrangement of reflecting hyperplanes for W. We show that they are self-adjoint and positive semidefinite. via two explicit factorizations into a symmetrized form A^t A. In one such factorization, A is a generalization of the projection of a simplex onto the linear ordering polytope. In the other factorization, A is the transition matrix for one of the well-studied Bidigare-Hanlon-Rockmore random walks on the chambers of an arrangement. We study the family of operators in which O is the conjugacy classes of Young subgroups of type (k,1^{n-k}). A special case within this family is the operator corresponding to random-to-random shuffling. We show in a purely enumerative fashion that these operators pairwise commute. We furthermore conjecture that they have integer spectrum, generalizing a conjecture of Uyemura-Reyes for the case k=n-1. We use representation theory to show that if O is a conjugacy class of rank one parabolics in W, the corresponding operator has integer spectrum. Our proof makes use of an (apparently) new family of twisted Gelfand pairs for W. We also study the family of operators in which O is the conjugacy classes of Young subgroups of type (2^k,1^{n-2k}). Here the construction of a Gelfand model for the symmetric group shows that these operators pairwise commute and that they have integer spectrum. For the symmetric group, we conjecture that apart from the two commuting families above, no other pair of operators of this form commutes.

We study partitions of the symmetric group which have desirable geometric properties. The statistical tests defined by such partitions involve counting all permutations in the equivalence classes. These permutations are the linear extensions of partially ordered sets specified by the data. Our methods refine rank tests of non-parametric statistics, such as the sign test and the runs test, and are useful for the exploratory analysis of ordinal data. Convex rank tests correspond to probabilistic conditional independence structures known as semi-graphoids. Submodular rank tests are classified by the faces of the cone of submodular functions, or by Minkowski summands of the permutohedron. We enumerate all small instances of such rank tests. Graphical tests correspond to both graphical models and to graph associahedra, and they have excellent statistical and algorithmic properties.

The notion of matroid has been generalized to Coxeter matroid by Gelfand and Serganova. To each pair (W, P) consisting of a finite irreducible Coxeter group W and parabolic subgroup P is associated a collection of objects called Coxeter matroids. The (ordinary) matroids are the special case where W is the symmetric group (the An case) and P is a maximal parabolic subgroup.This generalization of matroid introduces interesting combinatorial structures corresponding to each of the finite Coxeter groups. Borovik, Gelfand and White began an investigation of the Bn case, called symplectic matroids. This paper initiates the study of the Dn case, called orthogonal matroids. The main result (Theorem 2) gives three characterizations of orthogonal matroid: algebraic, geometric, and combinatorial. This result relies on a combinatorial description of the Bruhat order on Dn (Theorem 1). The representation of orthogonal matroids by way of totally isotropic subspaces of a classical orthogonal space...

It is well known that the moduli space of flat connections on a trivial principal bundle MxG, where G is a connected Lie group, is isomorphic to the representation variety Hom(\pi_1(M), G)/G. For a tiling T, viewed as a marked copy of R^d, we define a new kind of bundle called pattern equivariant bundle over T and consider the set of all such bundles. This is a topological invariant of the tiling space induced by T, which we call PREP(T), and we show that it is isomorphic to the direct limit lim_{f_n} Hom(\pi_1(\Gamma_n), G)/G, where \Gamma_n are the approximants to the tiling space and f_n are maps between them. G can be any group. As an example, we choose G to be the symmetric group S_3 and we calculate this direct limit for the Period Doubling tiling and its double cover, the Thue-Morse tiling, obtaining different results. This is the simplest topological invariant that can distinguish these two examples.

The symmetric group $S_3$ acts on $S^2 \times S^2 \times S^2$ by coordinate permutation, and the quotient space $(S^2 \times S^2 \times S^2)/S_3$ is homeomorphic to the complex projective space $\CC P^3$. In this paper, we construct an 124-vertex simplicial subdivision $(S^2 \times S^2 \times S^2)_{124}$ of the 64-vertex standard cellulation $S^2_4 \times S^2_4 \times S^2_4$ of $S^2 \times S^2 \times S^2$, such that the $S_3$-action on this cellulation naturally extends to an action on $(S^2 \times S^2 \times S^2)_{124}$. Further, the $S_3$-action on $(S^2 \times S^2 \times S^2)_{124}$ is "good", so that the quotient simplicial complex $(S^2 \times S^2 \times S^2)_{124}/S_3$ is a 30-vertex triangulation $\CC P^3_{30}$ of $\CC P^3$. In other words, we construct a simplicial realization $(S^2 \times S^2 \times S^2)_{124} \to \CC P^3_{30}$ of the branched covering $S^2 \times S^2 \times S^2 \to \CC P^3$. Finally, we apply the BISTELLAR program of Lutz on $\CC P^3_{30}$, resulting in an 18-vertex 2-neighbourly triangulation $\CC P^3_{18}$ of $\CC P^3$. The automorphism group of $\CC P^3_{18}$ is trivial. It may be recalled that, by a result of Arnoux and Marin, any triangulation of $\CC P^3$ requires at least 17 vertices. So, $\CC P^3_{18}$ is close to vertex-minimal, if not actually vertex-minimal. Moreover, no explicit triangulation of $\CC P^3$ was known so far.

Vertices of the 4-dimensional semi-regular polytope, snub 24-cell   and its symmetry group (W(D4)/C2):S3(W(D4)/C2):S3 of order 576 are represented in terms of quaternions with unit norm. It follows from the icosian representation of E8E8 root system. A simple method is employed to construct the E8E8 root system in terms of icosians which decomposes into two copies of the quaternionic root system of the Coxeter group W(H4)W(H4), while one set is the elements of the binary icosahedral group the other set is a scaled copy of the first. The quaternionic root system of H4H4 splits as the vertices of 24-cell and the snub 24-cell under the symmetry group of the snub 24-cell which is one of the maximal subgroups of the group W(H4)W(H4) as well as W(F4)W(F4). It is noted that the group is isomorphic to the semi-direct product of the proper rotation subgroup of the Weyl group of D4D4 with symmetric group of order 3 denoted by (W(D4)/C2):S3(W(D4)/C2):S3, the Coxeter notation for which is [3,4,3+][3,4,3+]. We analyze the vertex structure of the snub 24-cell and decompose the orbits of W(H4)W(H4) under the orbits of (W(D4)/C2):S3(W(D4)/C2):S3. The cell structure of the snub 24-cell has been explicitly analyzed with quaternions by using the subgroups of the group (W(D4)/C2):S3(W(D4)/C2):S3. In particular, it has been shown that the dual polytopes 600-cell with 120 vertices and 120-cell with 600 vertices decompose as 120=24+96120=24+96 and 600=24+96+192+288600=24+96+192+288 respectively under the group (W(D4)/C2):S3(W(D4)/C2):S3. The dual polytope of the snub 24-cell is explicitly constructed. Decompositions of the Archimedean W(H4)W(H4) polytopes under the symmetry of the group (W(D4)/C2):S3(W(D4)/C2):S3 are given in the appendix.