Moufang Quadrangles

Quadrangles embedded in metasymplectic spaces

This chapter presents various results about quadratic forms over a field complete with respect to a discrete valuation. The discussion is based on the assumption that K is a field of arbitrary characteristic which is complete with respect to a discrete valuation ν‎ and uses the usual convention that ν‎(0) = infinity. The chapter starts with a notation regarding the ring of integers of K and the natural map from it to the residue field, followed by a number of propositions regarding an anisotropic quadratic space. These include an anisotropic quadratic space with residual quadratic spaces, an unramified quadratic space of finite dimension, unramified finite-dimensional anisotropic quadratic forms over K, unramified anisotropic quadratic forms and a bilinear form, and a round quadratic space over K. The chapter concludes with a theorem that there exists an anisotropic quadratic form over K, unique up to isometry, and is non-singular.

This chapter investigates the consequences of the assumption that one Moufang set is weakly isomorphic to another. It first introduces some well-known facts about involutions which are assembled in a few lemmas, including those dealing with an involutory set, a biquaternion division algebra, and a quaternion division algebra with a standard involution. It then presents a notation for a non-trivial anisotropic quadratic space and another for an involutory set are presented, along with assumptions for a pointed anisotropic quadratic space and the standard involution of a quaternion. It also makes a number of propositions regarding the standard involution of a quaternion and a biquaternion. Results about weak isomorphisms between Moufang sets arising from involutory sets are given.

Remark. Theorem I is not true for some non-abelian collineation groups. For example, the non-abelian group G = PGL(3, q) acting naturally on PG(2, q3) (or on the Figueroa plane of order q3) has a point orbit P and a line orbit ℒ satisfying: The point orbit (P is the set of all points which are incident with exactly one line of a ^-invariant order q subplane and the line orbit L is the set of lines incident with no point of this sub-plane. See, for example, Brown (1983, §6).

In this paper the stability of the quadratic equation is considered on arbitrary groups. Since the quadratic equation is stable on Abelian groups, this paper examines the stability of the quadratic equation on noncommutative groups. It is shown that the quadratic equation is stable on $n$-Abelian groups when $n$ is a positive integer. The stability of the quadratic equation is also established on the noncommutative group $T(2, K)$, where $K$ is an arbitrary commutative field. It is proved that every group can be embedded into a group in which the quadratic equation is stable.

The general solution of the functional equation f(xy) = f(x)h(y) + f(y) on abelian groups is well-known. We present methods for solving this equation on various non-abelian groups. In particular we treat the equation on semidirect products, then extend this treatment to solvable groups. We also find the solution on perfect groups. These results also apply to the more general equation f(xy) = g(x)h(y) + k(y).

The Cramer-Shoup cryptosystem has attracted much attention from the research community, mainly due to its efficiency in encryption/decryption, as well as the provable reductions of security against adaptively chosen ciphertext attacks in the standard model. At TCC 2005, Vasco et al. proposed a method for building Cramer-Shoup like cryptosystem over non-abelian groups and raised an open problem for finding a secure instantiation. Based on this work, we present another general framework for constructing Cramer-Shoup like cryptosystems. We firstly propose the concept of index exchangeable family (IEF) and an abstract construction of Cramer-Shoup like encryption scheme over IEF. The concrete instantiations of IEF are then derived from some reasonable hardness assumptions over abelian groups as well as non-abelian groups, respectively. These instantiations ultimately lead to simple yet efficient constructions of Cramer-Shoup like cryptosystems, including new non-abelian analogies that can be potential solutions to Vasco et al.'s open problem. Moreover, we propose a secure outsourcing method for the encryption of the non-abelian analog based on the factorization problem over non-commutative groups. The experiments clearly indicate that the computational cost of our outsourcing scheme can be significantly reduced thanks to the load sharing with cloud datacenter servers.

In the framework of constructing mirror symmetric pairs of Calabi-Yau manifolds, P. Berglund, T. Hübsch and M. Henningson considered a pair $(f,G)$ consisting of an invertible polynomial $f$ and a finite abelian group $G$ of its diagonal symmetries and associated to this pair a dual pair $(\widetilde{f}, \widetilde{G})$. A. Takahashi suggested a generalization of this construction to pairs $(f, G)$ where $G$ is a non-abelian group generated by some diagonal symmetries and some permutations of variables. In a previous paper, the authors showed that some mirror symmetry phenomena appear only under a special condition on the action of the group $G$: a parity condition. Here we consider the orbifold Euler characteristic of the Milnor fibre of a pair $(f,G)$. We show that, for an abelian group $G$, the mirror symmetry of the orbifold Euler characteristics can be derived from the corresponding result about the equivariant Euler characteristics. For non-abelian symmetry groups we show that the orbifold Euler characteristics of certain extremal orbit spaces of the group $G$ and the dual group $\widetilde{G}$ coincide. From this we derive that the orbifold Euler characteristics of the Milnor fibres of dual periodic loop polynomials coincide up to sign.

This chapter deals with the case that the building at infinity Λ‎ of the Bruhat-Tits building Ξ‎ is a Moufang quadrangle of type F₄. It begins with the hypothesis stating that Λ‎ = (K, L, q) is a quadratic space of type F₄, K is complete with respect to a discrete valuation ν‎ and F is closed with respect to ν‎, Λ‎ is the Moufang quadrangle corresponding to a root group sequence, and R₀ and R₁ as the two residues of Ξ‎. The chapter also considers the theorem supposing that Λ‎ is of type F₄ and that R₀ and R₁ are not both indifferent, and claims that both cases really occur. Finally, it presents the proposition that R₀ and R₁ are both indifferent if and only if q is totally wild.

In this paper, be the non-abelian metabelian group and denotes the automorphism group of group . A group is metabelian group if its commutator group is abelian or if it has an abelian normal subgroupin which the factor group is also abelian. There are 35 non-Abelian metabelian groups of order less than or equal to 24. In this study, we investigate the automorphism groups of all non-abelian metabelian groups of order less than or equal to 24 and the verification has been made through GAP (Groups Algorithm Programming) software.

A new method for the construction of the binary quantum stabilizer codes is provided, where the construction is based on Abelian and non-Abelian groups association schemes. The association schemes based on non-Abelian groups are constructed by bases for the regular representation from U6n, T4n, V8n and dihedral D2n groups. By using Abelian group association schemes followed by cyclic groups and non-Abelian group association schemes a list of binary stabilizer codes up to 40 qubits is given in tables 4, 5 and 10. Moreover, several binary stabilizer codes of minimum distances 5, 7 and 8 with good quantum parameters is presented. The preference of this method specially for Abelian group association schemes is that one can construct any binary quantum stabilizer code with any distance by using the commutative structure of association schemes.

The research of my dissertation was motivated by the conjecture of Thomassen that every 4-connected line graph is hamiltonian and by the conjecture of Matthews and Sumner that every 4-connected claw-free graph is hamiltonian. Towards the hamiltonian line graph problem, we proved that every 3-edge-connected, essentially 4-edge-connected graph G has a spanning eulerian subgraph, if for every pair of adjacent vertices u and v, dG(u) + dG(v) ≥ 9. A straight forward corollary is that every 4-connected, essentially 6-connected line graph with minimum degree at least 7 is hamiltonian.;We also investigate graphs G such that the line graph L(G) is hamiltonian connected when L( G) is 4-connected. Ryjacek and Vrana recently further conjectured that every 4-connected line graph is hamiltonian-connected. In 2001, Kriesell proved that every 4-connected line graph of a claw free graph is hamiltonian connected. Recently, Lai et al showed that every 4-connected line graph of a quasi claw free graph is hamiltonian connected, and that every 4-connected line graph of an almost claw free graph is hamiltonian connected. In 2009, Broersma and Vumer discovered the P3-dominating (P3D) graphs as a superfamily that properly contains all quasi claw free graphs, and in particular, all claw-free graphs. Here we prove that every 4-connected line graph of a P3D graph is hamiltonian connected, which extends several former results in this area.;R. Gould [15] asked what natural graph properties of G and H are sufficient to imply that the product of G and H is hamiltonian. We first investigate the sufficient and necessary conditions for G x H being hamiltonian or traceable when G is a hamiltonian graph and H is a tree. Then we further investigate sufficient and necessary conditions for G x H being hamiltonian connected, or edge-pancyclic, or pan-connected.;The problem of group colorings of graphs is also investigated in this dissertation. Group coloring was first introduced by Jeager et al. [21]. They introduced a concept of group connectivity as a generalization of nowhere-zero flows. They also introduced group coloring as a dual concept to group connectivity. Prior research on group chromatic number was restricted to simple graphs, and considered only Abelian groups in the definition of chi g(G). The behavior of group coloring for multigraphs is different to that of simple graphs. Thus we extend the definition of group coloring by considering general groups (both Abelian groups and non-Abelian groups), and investigate the properties of chig for multigraphs

The general solution of the multiplicative symmetric functional equation $$f(f(x)y)=f(xf(y))$$, in abelian groups, was given by Dhombres. But it is still an open problem in non-abelian groups and abelian quasi-groups. In this paper, by using a new approach introduced by the author, we give a vast class of its solutions in arbitrary semigroups. It also works for non-abelian groups, and gives its general solution in abelian groups with a different representation from Dhombres’s solution. Also, we introduce some related equations and closed relationships between decomposer type and multiplicative symmetric equations. It is worth noting that the new approach and methods can be used for studying the equation on magmas (groupoids) as the most general algebraic structures for the topic. At last we pose a conjecture that guesses the general solution in arbitrary groups (and possibly semigroups).

Some results from [2], [5], [6] are generalized for finite automata over arbitrary groups. The accepting power is smaller when abelian groups are considered, in comparison with the non-abelian groups. We prove that this is due to the commutativity. Each language accepted by a finite automaton over an abelian group is actually a unordered vector language. Finally, deterministic finite automata over groups are investigated.

Addition sets in a finite group