On Finite Elation Generalized Quadrangles with Symmetries

In the 1980's, the work of Frenkel, Lepowsky and Meurman, along with that of Borcherds, culminated in the notion of vertex operator algebra, and an example whose full symmetry group is the largest sporadic simple group: the Monster. Thus it was shown that the vertex operators of mathematical physics play a role in finite group theory. In this article we describe an extension of this phenomenon by introducing the notion of enhanced vertex operator algebra, and constructing examples that realize other sporadic simple groups, including one that is not involved in the Monster.

Probability that the commutator of two group elements is equal to a given element

For a formation 𝔉 of finite groups, consider the graph whose vertices are elements of a finite group, with two vertices connected by an edge if and only if they generate a non-𝔉-group as elements of the group. A hereditary formation 𝔉 is said to be regular if the set of all isolated vertices of the described graph coincides with the intersection of all 𝔉-maximal subgroups in every group. In this paper, a constructive description of saturated regular formations of soluble groups is given, which improves the results of Lucchini and Nemmi. In particular, it is shown that saturated regular non-empty formations of soluble groups are just hereditary formations 𝔉 of soluble groups that contain every group whose cyclic primary subgroups are all 𝐾-𝔉-subnormal. Also, we prove that the lattice of saturated regular formations of soluble groups is lattice isomorphic to the Steinitz lattice.

A crystallographic bar-joint framework, C in Rd, is shown to be almost periodically infinitesimally rigid if and only if it is strictly periodically infinitesimally rigid and the rigid unit mode (RUM) spectrum, Ω (C), is a singleton. Moreover, the almost periodic infinitesimal flexes of C are characterised in terms of a matrix-valued function, ΦC(z), on the d-torus, Td, determined by a full rank translation symmetry group and an associated motif of joints and bars.

New constraints on resonance saturation in chiral perturbation theory are investigated. These constraints arise because each consistent saturation scheme must map to a representation of the full QCD chiral symmetry group. The low-energy constants of chiral perturbation theory are then related by a set of mixing angles. It is shown that vector meson dominance is a consequence of the fact that nature has chosen the lowest-dimensional nontrivial chiral representation. It is further shown that chiral symmetry places an upper bound on the mass of the lightest scalar in the hadron spectrum.

The vanishing phase space generator of the full four-dimensional diffeomorphism-related symmetry group in the context of the Barbero–Immirz–Holst Lagrangian is derived directly, for the first time, from Noether’s second theorem. Its applicability in the construction of classical diffeomorphism invariants is reviewed.

Geometry of the solution space of the self-dual Yang–Mills (SDYM) equations in Euclidean four-dimensional space is studied. Combining the twistor and group-theoretic approaches, we describe the full infinite-dimensional symmetry group of the SDYM equations and its action on the space of local solutions to the field equations. It is argued that owing to the relation to a holomorphic analogue of the Chern–Simons theory, the SDYM theory may be as solvable as 2D rational conformal field theories, and successful nonperturbative quantization may be developed. An algebra acting on the space of self-dual conformal structures on a 4-space (an analogue of the Virasoro algebra) and an algebra acting on the space of self-dual connections (an analogue of affine Lie algebras) are described. Relations to problems of topological and N=2 strings are briefly discussed.

Machine-learning force fields (MLFF) should be accurate, computationally and data efficient, and applicable to molecules, materials, and interfaces thereof. Currently, MLFFs often introduce tradeoffs that restrict their practical applicability to small subsets of chemical space or require exhaustive datasets for training. Here, we introduce the Bravais-Inspired Gradient-Domain Machine Learning (BIGDML) approach and demonstrate its ability to construct reliable force fields using a training set with just 10–200 geometries for materials including pristine and defect-containing 2D and 3D semiconductors and metals, as well as chemisorbed and physisorbed atomic and molecular adsorbates on surfaces. The BIGDML model employs the full relevant symmetry group for a given material, does not assume artificial atom types or localization of atomic interactions and exhibits high data efficiency and state-of-the-art energy accuracies (errors substantially below 1 meV per atom) for an extended set of materials. Extensive path-integral molecular dynamics carried out with BIGDML models demonstrate the counterintuitive localization of benzene–graphene dynamics induced by nuclear quantum effects and their strong contributions to the hydrogen diffusion coefficient in a Pd crystal for a wide range of temperatures.

We study an equifacetted self dual 3-sphere S McM of Peter McMullen, [10], in particular its automorphism group A(S McM ) and its relation to the Coxeter group H 4 of the 600-cell. A closely related equifacetted polyhedral 3-sphere (240-cell) with 240 facets and 120 vertices has the same automorphism group. Both these 3-spheres and the polar dual of the last one cannot occur as the boundary complex of a (convex) 4-polytope with A(S McM ) as their full Euclidean symmetry. It is an open problem, whether there exist one of these three 4-polytopes at all. Their combinatorial symmetry would differ from their Euclidean one within their whole realization space, similar to the example given in [3], see also [2]. Tackling these problems with methods from computational synthetic geometry [5] fail because of the large problem size. Therefore, a partial Euclidean symmetry assumption for the questionable polytope is natural. On the other hand, we show that even a certain subgroup of order 5 of the full combinatorial symmetry group A(S McM ) of order 1200 cannot occur as a Euclidean symmetry for McMullen's questionable polytope.

Let g be a fixed element of a finite group G. We introduce the g-noncommuting graph of G whose vertex set is whole elements of the group G and two vertices x,y are adjacent whenever [x,y] g and [y,x] g. We denote this graph by . In this paper, we present some graph theoretical properties of g-noncommuting graph. Specially, we investigate about its planarity and regularity, its clique number and dominating number. We prove that if G, H are isoclinic groups with |Z (G)|=|Z (H)|, then their associated graphs are isomorphic.

We consider various generalizations of the effective exponentiation problem, also known as the addition chains problem in additive terminology. We attempt to evaluate and consistently compare the results and the methods that were used to obtain them. The considered generalizations are complexity problems (i.e. finding the minimum number of required operations) for the computation a monomial of many variables, a set of powers, elements of finite Abelian groups, elements of free semigroups, systems of monomials, systems of integer linear forms, systems of elements of free Abelian groups along with some other problems. For all these directions of research we overview both the classical results and the results of the author and his followers.

A blackbox secret sharing (BBSS) scheme works in exactly the same way for all finite Abelian groups G; it can be instantiated for any such group G and only black-box access to its group operations and to random group elements is required. A secret is a single group element and each of the n players’ shares is a vector of such elements. Share-computation and secret-reconstruction is by integer linear combinations. These do not depend on G, and neither do the privacy and reconstruction parameters t, r. This classical, fundamental primitive was introduced by Desmedt and Frankel (CRYPTO 1989) in their context of “threshold cryptography.” The expansion factor is the total number of group elements in a full sharing divided by n. For threshold BBSS with t-privacy (\(1\le t \le n-1\)), \(t+1\)-reconstruction and arbitrary n, constructions with minimal expansion \(O(\log n)\) exist (CRYPTO 2002, 2005).

1. In studying the structure of abstract groups, one is often confronted with the seemingly trivial question: when do we consider a group (or class of groups) as known? The question has many aspects, and is usually a very complex one. Of course, the group is formally known as soon as its multiplication table is determined; but in most cases, little further information can be obtained from the multiplication table itself on the structural elements of the group, such as subgroups, normal subgroups, automorphisms, etc. If one is however, interested in the classification of groups belonging to a certain class, then the problem is concentrated on the enumeration of groups having some common property, and the structural elements of the individual groups are of a secondary importance. In this case, the problem can be formulated in the following way: Find some finite algebraical procedure, depending on some integral or more general algebraical parameters, which enables us to construct, in an abstract or symbolic way, each group belonging to the class under consideration. A group is of course constructed if its elements are known, and the groupproduct of any two elements can be calculated by means of the defining procedure. It is desirable that the parameters, called invariants of the group, should be characteristic of the group in the sense that no two distinct systems of operations should define the same group. A classical example for such an interpretation of the structure problem is furnished by the fundamental theorem of finite Abelian groups. In this paper we are concerned with the enumeration of certain finite metabelian groups. (Metabelian is a group with Abelian commutator subgroup.) The problem in all its generality is a very difficult one, and far from being solved in the above sense. However, special classes of metabelian groups are accessible to investigation, and such investigations have an interest in so far as they clear more aspects of the general problem. In studying the structure of metabelian groups, the method of group extensions is often found to be the most natural tool. As an extensive use will be made of this method in this paper, I want to formulate it in such a way as it seems most suitable for the case discussed here. A similar but formally different formulation was given by A. Scholz1. See also W. Magnus2. Let ( be a finite metabelian group, additively written, 2t an Abelian normal subgroup of (M such that the quotient group e = (5/Wt is Abelian. Elements

Finite Affine Groups: Cycle Indices, Hall–Littlewood Polynomials, and Probabilistic Algorithms

Given a finite group G, the solubilizer of an element x, denoted by Sol G ( x ) , is the set of all elements y such that 〈 x , y 〉 is a soluble subgroup of G. In this paper, we provide a classification for all solubilizers of elements in minimal simple groups. We also examine these sets to explore their properties by discussing some computational methods and making some conjectures for further work.