Element orders in coverings of symmetric and alternating groups

We prove that if the factor group H=G/N of a finite group G is isomorphic to a symmetric or alternating group of degree m, where m≥5 and N≠1, then G has an element whose order is distinct from any element’s order in H. Supported by RFFR grant No. 96-01-01893. Translated fromAlgebra i Logika, Vol. 38, No. 3, pp. 296–315, May–June, 1999.     

The fine structures of three symmetric Latin squares with even orders

Denote by SFin(v) the set of all integer pairs (t, s) for which there exist three symmetric Latin squares of order v on the same set having fine structure (t, s). We completely determine the set SFin(2n) for any integer n ≥ 5.     

Generalized symmetric planes

In this paper we relate the theory of stable planes to the theory of generalized symmetric spaces in the sense of differential geometry where the symmetries may be of arbitrary order. This leads to the notion of a generalized symmetric plane. We assign to every generalized symmetric plane an associated infinitesimal model and show that the associated infinitesimal model essentially determines a generalized symmetric plane up to global isomorphism. In particular, every generalized symmetric plane with an abelian group of transvections is a topological translation plane.     

Eigenvalues and eigenvectors of symmetric centrosymmetric matrices

It is proved that the eigenvectors of a symmetric centrosymmetric matrix of order N are either symmetric or skew symmetric, and that there are ?N/2? symmetric and ?N/2? skew symmetric eigenvectors. Some previously known but widely scattered facts about symmetric centrosymmetric matrices are presented for completeness. Special cases are considered, in particular tridiagonal matrices of both odd and even order, for which it is shown that the eigenvectors corresponding to the eigenvalues arranged in descending order are alternately symmetric and skew symmetric provided the eigenvalues are distinct.     

On generalized symmetric Finsler spaces

In this paper, we study generalized symmetric Finsler spaces. We first study some existence theorems, then we consider their geometric properties and prove that any such space can be written as a coset space of a Lie group with an invariant Finsler metric. Finally we show that each generalized symmetric Finsler space is of finite order and those of even order reduce to symmetric Finsler spaces and hence are Berwaldian.     

Embedding partial totally symmetric quasigroups

This paper concerns the embedding problem for partial totally symmetric quasigroups. For all n?9, it is shown that any partial totally symmetric quasigroup of order n can be embedded in a totally symmetric quasigroup of order v if v is even and v?2n+4, and this is the best possible such inequality.     

s-Sequences and symmetric algebras

In this paper we introduce the concept of s-sequences in order to study the symmetric algebra of a module. It is explained how s-sequences are related to d-sequences. The theory is applied to study the symmetric algebra of generic modules. We also compute the Gr?bner basis of the defining ideals of these symmetric algebras. Received: 12 September 2000 / Accepted: 26 January 2001     

Relative symmetric polynomials

In this article, we introduce the notion of a relative symmetric polynomial with respect to a permutation group and an irreducible character and we give answers for some natural questions about their structures. In order to study symmetric polynomials with respect to linear characters, we define the concept of relative Vandermonde polynomial. Finally, we present some interesting research problems concerning relative symmetric polynomials.     

On weakly symmetric graphs of order twice a prime

A graph is weakly symmetric if its automorphism group is both vertex-transitive and edge-transitive. In 1971, Chao characterized all weakly symmetric graphs of prime order and showed that such graphs are also transitive on directed edges. In this paper we determine all weakly symmetric graphs of order twice a prime and show that these graphs too are directed-edge transitive.     

On embedding incomplete symmetric latin squares

Necessary and sufficient conditions are obtained for the extendibility of an r × r symmetric Latin rectangle to an n × n symmetric Latin square. These conditions imply that any incomplete n × n symmetric Latin square can be embedded in a complete symmetric Latin square of order 2n. Also, any incomplete n × n symmetric diagonal Latin square can be embedded in a complete symmetric diagonal Latin square of order 2n + 1.     

Nondifferentiable second order symmetric duality in multiobjective programming

A pair of Mond–Weir type nondifferentiable multiobjective second order symmetric dual programs is formulated and symmetric duality theorems are established under the assumptions of second order F-pseudoconvexity/F-pseudoconcavity.     

An eigenvalue problem for a symmetric Toeplitz matrix

An algorithm is developed which determines eigenvalues for a symmetric Toeplitz matrix. To this end, we substantiate the generality of eigenvalues problems for a symmetric Toeplitz matrix and for a persymmetric Hankel one. The latter is reduced to an eigenvalue problem for a persymmetric Jacobi matrix. In the even order case, the problem reduces to a Jacobi matrix with halved order.     

A basic class of symmetric orthogonal polynomials using the extended Sturm-Liouville theorem for symmetric functions

In this research, by applying the extended Sturm-Liouville theorem for symmetric functions, a basic class of symmetric orthogonal polynomials (BCSOP) with four free parameters is introduced and all its standard properties, such as a generic second order differential equation along with its explicit polynomial solution, a generic orthogonality relation, a generic three term recurrence relation and so on, are presented. Then, it is shown that four main sequences of symmetric orthogonal polynomials can essentially be extracted from the introduced class. They are respectively the generalized ultraspherical polynomials, generalized Hermite polynomials and two other sequences of symmetric polynomials, which are finitely orthogonal on (−∞,∞) and can be expressed in terms of the mentioned class directly. In this way, two half-trigonometric sequences of orthogonal polynomials, as special sub-cases of BCSOP, are also introduced.     

Highly symmetric generalized circulant permutation matrices

In this paper we investigate generalized circulant permutation matrices of composite order. We give a complete characterization of the order and the structure of symmetric generalized k-circulant permutation matrices in terms of circulant and retrocirculant block (0, 1)-matrices in which each block contains exactly one or two entries 1. In particular, we prove that a generalized k-circulant matrix A of composite order n = km is symmetric if and only if either k = m − 1 or k ≡ 0 or k ≡ 1 mod m, and we obtain three basic symmetric generalized k-circulant permutation matrices, from which all others are obtained via permutations of the blocks or by direct sums. Furthermore, we extend the characterization of these matrices to centrosymmetric matrices.     

Non-symmetric perturbations of symmetric Dirichlet forms

We provide a path-space integral representation of the semigroup associated with the quadratic form obtained by lower order perturbation of a symmetric local Dirichlet form. The representation is a combination of Feynman-Kac and Girsanov formulas, and extends previously known results in the framework of symmetric diffusion processes through the use of the Hardy class of smooth measures, which contains the Kato class of smooth measures.     

On R*-sequenceability and symmetric harmoniousness of groups

We introduce a special harmoniousness called symmetric harmoniousness of groups and extend the R^*-sequenceability of abelian groups to nonabelian groups. We prove that the direct product of an R^*-sequenceable group of even order with a symmetric harmonious group of odd order is R^*-sequenceable. Examples of nonabelian R^*-sequenceable groups and nonabelian symmetric harmonious groups are given. It is shown that the nonabelian groups of order 3q (q prime) are symmetric harmonious. © 1994 John Wiley & Sons, Inc.     

Linear orders and semiorders close to an interval order

The purpose of this paper is to study the properties of the linear orders and semiorders at minimum symmetric difference distance from a given interval order on a finite set.     

Pentavalent symmetric graphs of order 2p3

A graph is symmetric or 1-regular if its automorphism group is transitive or regular on the arc set of the graph, respectively. We classify the connected pentavalent symmetric graphs of order 2p~3 for each prime p. All those symmetric graphs appear as normal Cayley graphs on some groups of order 2p~3 and their automorphism groups are determined. For p = 3, no connected pentavalent symmetric graphs of order 2p~3 exist. However, for p = 2 or 5, such symmetric graph exists uniquely in each case. For p 7, the connected pentavalent symmetric graphs of order 2p~3 are all regular covers of the dipole Dip5 with covering transposition groups of order p~3, and they consist of seven infinite families; six of them are 1-regular and exist if and only if 5 |(p- 1), while the other one is 1-transitive but not 1-regular and exists if and only if 5 |(p ± 1). In the seven infinite families, each graph is unique for a given order.     

Extreme symmetric doubly stochastic matrices

We characterize the extreme points of the polytope of symmetric doubly stochastic matrices of a given arbitrary order.     

Asymptotically symmetric embeddings and symmetric quasicircles

A well-known characterization of quasicircles is the following: A Jordan curve in the complex plane is a quasicircle if and only if it is the image of the unit circle under a quasisymmetric embedding. In this paper we try to characterize a subclass of quasicircles, namely, symmetric quasicircles, by introducing the concept of asymptotically symmetric embeddings. We show that a Jordan curve in the complex plane is a symmetric quasicircle if and only if it is the image of the unit circle under an asymptotically symmetric embedding.