Related papers: Configuration sets for groups
In this paper the structure of the Cayley graphs and G-graphs of some gyro-groups are studied and some properties of them will be proved. Moreover we review some special gyro-groups including: gyro-commutative gyrogroups, dihedral…
Given a finite abelian group $G$ and elements $x, y \in G$, we prove that there exists $\phi \in \text{Aut}(G)$ such that $\phi(x) = y$ if and only if $G/\langle x \rangle \cong G/\langle y \rangle$. This result leads to our development of…
We review properties of affine special Kaehler structures focusing on singularities of such structures in the simplest case of real dimension two. We describe all possible isolated singularities and compute the monodromy of the flat…
New criteria for which Cayley graphs of cyclic groups of any order can be completely determined--up to isomorphism--by the eigenvalues of their adjacency matrices is presented. Secondly, a new construction for pairs of nonisomorphic Cayley…
We show that whether loops can be shortcut in a group's Cayley graph depends on the choice of finite generating set. Our example is the direct product of two rank-2 free groups and a consequence is that this group has asymptotic cones with…
For a finite noncyclic group $G$, let $\Cyc(G)$ be a set of elements $a$ of $G$ such that $\langle a,b\rangle$ is cyclic for each $b$ of $G$. The noncyclic graph of $G$ is a graph with the vertex set $G\setminus \Cyc(G)$, having an edge…
We generalise the standard constructions of a Cayley graph in terms of a group presentation by allowing some vertices to obey different relators than others. The resulting notion of presentation allows us to represent every vertex…
This article studies automorphism groups of graph products of arbitrary groups. We completely characterise automorphisms that preserve the set of conjugacy classes of vertex groups as those automorphisms that can be decomposed as a product…
Associated to each finite dimensional linear representation of a group G, there is a vector bundle over the classifying space BG. This construction was studied extensively for compact groups by Atiyah and Segal. We introduce a homotopy…
We present a general diagrammatic approach to the construction of efficient algorithms for computing the Fourier transform of a function on a finite group. By extending work which connects Bratteli diagrams to the construction of Fast…
Let M be a simply-connected closed manifold and consider the (ordered) configuration space of $k$ points in M, F(M,k). In this paper we construct a commutative differential graded algebra which is a potential candidate for a model of the…
We introduce a remarkable subset "the stem" of the set of positive roots of a reduced root system. The stem determines several interesting decompositions of the corresponding reductive Lie algebra. It gives also a nice simple three…
We present several results regarding the connectivity of McKay quivers of finite-dimensional complex representations of finite groups, with no restriction on the faithfulness or self-duality of the representations. We give examples of McKay…
We start by introducing the basics of configurations of points and lines, and then move into discussing symmetry groups of these configurations. Specifically, we explore how we might classify the symmetries of $(9_3)$ and $(10_3)$ geometric…
In the present paper we define homogeneous algebraic systems. Particular cases of these systems are: semigroup (monoid, group) system. These algebraic systems were studied by J. Loday, A. Zhuchok, T. Pirashvili, N. Koreshkov. Quandle…
A group $G$ has a Frobenius graphical representation (GFR) if there is a simple graph $\varGamma$ whose full automorphism group is isomorphic to $G$ and it acts on vertices as a Frobenius group. In particular, any group $G$ with GFR is a…
A number of authors have studied the question of when a graph can be represented as a Cayley graph on more than one nonisomorphic group. The work to date has focussed on a few special situations: when the groups are $p$-groups; when the…
The paper proposes a new and systematic approach to the so-called black box group methods in computational group theory. As the starting point of our programme, we construct Frobenius maps on black box groups of untwisted Lie type in odd…
Signed difference sets have interesting applications in communications and coding theory. A $(v,k,\lambda)$-difference set in a finite group $G$ of order $v$ is a subset $D$ of $G$ with $k$ distinct elements such that the expressions…
A Cayley graph for a group $G$ is CCA if every automorphism of the graph that preserves the edge-orbits under the regular representation of $G$ is an element of the normaliser of $G$. A group $G$ is then said to be CCA if every connected…