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A permutation φ on a group G is called a skew morphism of G if φ (1) = 1 , and there exists an integer-valued function π : G → Z m , where m is the order of φ , such that φ (a b) = φ (a) φ π (a) (b) , for all a , b ∈ G . A skew morphism φ is smooth if the associated power function π of φ takes constant values on each orbit of φ . In this paper, we shall classify the smooth skew morphisms of semi-dihedral groups. [ABSTRACT FROM AUTHOR]

AbstractA finite group G is an exact product of two subgroups A and B if G = AB and A∩B={1G}. In this paper, for all odd numbers k≥3, using the theory of skew morphisms and associated extended power functions, we present a classification of exact products of a cyclic group and a dihedral group D2k. As an application the regular generalized Cayley maps of odd valency over cyclic groups are classified. [ABSTRACT FROM AUTHOR]

Starting from the very first principles we derive explicit parameterizations of the ortho-symplectic matrices in the real four-dimensional Euclidean space. These matrices depend on a set of four real parameters which splits naturally as a union of the real line and the three-dimensional space. It turns out that each of these sets is associated with a separate Lie algebra which after exponentiations generates Lie groups that commute between themselves. Besides, by making use of the Cayley and Fedorov maps, we have arrived at alternative realizations of the ortho-symplectic matrices in four dimensions. Finally, relying on the fundamental structure results in Lie group theory we have derived one more explicit parameterization of these matrices which suggests that the obtained earlier results can be viewed as a universal method for building the representations of the unitary groups in arbitrary dimension. [ABSTRACT FROM AUTHOR]

Starting from the very first principles we derive explicit parameterizations of the ortho-symplectic matrices in the real four-dimensional Euclidean space. These matrices depend on a set of four real parameters which splits naturally as a union of the real line and the three-dimensional space. It turns out that each of these sets is associated with a separate Lie algebra which after exponentiations generates Lie groups that commute between themselves. Besides, by making use of the Cayley and Fedorov maps, we have arrived at alternative realizations of the ortho-symplectic matrices in four dimensions. Finally, relying on the fundamental structure results in Lie group theory we have derived one more explicit parameterization of these matrices which suggests that the obtained earlier results can be viewed as a universal method for building the representations of the unitary groups in arbitrary dimension.

In this paper, we determine elliptical motions that occur on any given ellipsoid in 3D space without using affine transformations. To this end, first, we define the generalized Euclidean inner product whose sphere is the given ellipsoid, and determine skew symmetric matrices and the generalized vector product related to the 3D generalized Euclidean inner product space. Finally, we generate elliptical rotation matrices in 3D generalized Euclidean space using the famous Rodrigues, Cayley, and Householder methods. The formulas and results obtained are supported with numerical examples. We also give an algorithm for generalized elliptical rotation. [ABSTRACT FROM AUTHOR]

A skew morphism of a finite group A is a permutation φ on A fixing the identity element of A, and for which there exists an integer-valued function π : A → Z | φ | on A such that φ (a b) = φ (a) φ π (a) (b) for all a , b ∈ A . Moreover, the period of φ is the smallest positive integer d such that π (φ d (a)) ≡ π (a) (mod | φ |) for all a ∈ A . In the case where d = 1 , the skew morphism φ is called smooth. It is well known that if φ is a skew morphism of period d, then φ d is a smooth skew morphism. Thus, every skew morphism of period d may be extracted as a dth root of a smooth skew morphism. In this paper, we introduce a new concept of average function to investigate skew morphisms and as an application we present a classification of smooth skew morphisms of the dicyclic groups. [ABSTRACT FROM AUTHOR]

An orientably-regular map M is a 2-cell embedding of a finite connected graph in a closed orientable surface such that the group Aut ∘ M of orientation-preserving automorphisms of M acts transitively on the set of arcs. Such a map M is called a Cayley map for the finite group G if Aut ∘ M contains a subgroup, which is isomorphic to G and acts regularly on the set of vertices. Conder and Tucker (2014) classified the regular Cayley maps for finite cyclic groups, and obtain two two-parameter families M (n , r) , one for odd n and one for even n , where n is the order of the regular cyclic group and r is a positive integer satisfying certain arithmetical conditions. In this paper, we classify the regular Cayley maps for dihedral groups in the same fashion. Five two-parameter families M i (n , r) , 1 ≤ i ≤ 5 , are derived, where 2 n is the order of the regular dihedral group and r is an integer satisfying certain arithmetical conditions. For each map M i (n , r) , we determine its valence and covalence, and also describe the structure of the group Aut ∘ M i (n , r). Unlike the approach of Conder and Tucker, which is entirely algebraic, we follow the traditional combinatorial representation of Cayley maps, and use a combination of permutation group theoretical techniques, the method of quotient Cayley maps, and computations with skew morphisms. [ABSTRACT FROM AUTHOR]

Let G be a simply connected semisimple algebraic groups over ℂ and let ρ:G→GL(Vλ) be an irreducible representation of G of highest weight λ. Suppose that ρ has finite kernel. Springer defined an adjoint-invariant regular map with Zariski dense image from the group to the Lie algebra, 휃λ:G→픤, which depends on λ. This map, 휃λ, takes the maximal torus T of G to its Lie algebra 픱. Thus, for a given simple group G and an irreducible representation Vλ, one may write , where we take the simple coroots as a basis for 픱. We give a complete determination for these coefficients ci(t) for any simple group G as a sum over the weights of the torus action on Vλ. [ABSTRACT FROM AUTHOR]

A regular map M is a 2-cell embedding of a connected graph into an orientable surface such that the group of all orientation-preserving automorphisms of the embedding acts transitively on the set of all incident vertex-edge pairs called arcs. Such a map M is called a regular Cayley map for the finite group G if M is the embedding of a Cayley graph C ( G , S ) such that G induces a vertex-transitive group of map automorphisms preserving orientation. In addition, if there is an orientation-reversing automorphism, the map is called reflexible. In this paper, we classify all reflexible Cayley maps for dihedral groups. [ABSTRACT FROM AUTHOR]

This paper deals with the Cayley graph Cay ( Sym n , T n ) , where the generating set consists of all block transpositions. A motivation for the study of these particular Cayley graphs comes from current research in Bioinformatics. As the main result, we prove that Aut ( Cay ( Sym n , T n ) ) is the product of the left translation group and a dihedral group D n + 1 of order 2 ( n + 1 ) . The proof uses several properties of the subgraph Γ of Cay ( Sym n , T n ) induced by the set T n . In particular, Γ is a 2 ( n − 2 ) -regular graph whose automorphism group is D n + 1 , Γ has as many as n + 1 maximal cliques of size 2 , and its subgraph Γ ( V ) whose vertices are those in these cliques is a 3-regular, Hamiltonian, and vertex-transitive graph. A relation of the unique cyclic subgroup of D n + 1 of order n + 1 with regular Cayley maps on Sym n is also discussed. It is shown that the product of the left translation group and the latter group can be obtained as the automorphism group of a non- t -balanced regular Cayley map on Sym n . [ABSTRACT FROM AUTHOR]

In this paper, we determine non-parabolic conical motions that occur on any given ellipse or hyperbola without using affine transformations. To achieve this aim, first, we define a generalized inner product whose circle is the given ellipse or hyperbola, and then determine elliptical and hyperbolic versions of skew-symmetric and orthogonal matrices using the associated inner product. Finally, we generate elliptical and hyperbolic versions of rotation and reflection matrices using the famous Rodrigues, Cayley, and Householder transformations. For each of the generalized formulas, we give numerical examples. [ABSTRACT FROM AUTHOR]

We obtain matrix of the rotation about arbitrary lightlike axis in three-dimensional Minkowski space by deriving the Rodrigues' rotation formula and using the corresponding Cayley map. We prove that a unit timelike split quaternion q with a lightlike vector part determines rotation R about lightlike axis and show that a split quaternion product of two unit timelike split quaternions with null vector parts determines the rotation about a spacelike, a timelike or a lightlike axis. Finally, we give some examples. [ABSTRACT FROM AUTHOR]

A linear algebraic group G over a field K is called a Cayley K -group if it admits a Cayley map, i.e., a G -equivariant K -birational isomorphism between the group variety G and its Lie algebra. We classify real reductive algebraic groups of absolute rank 1 and 2 that are Cayley R -groups. [ABSTRACT FROM AUTHOR]

We use the term half-regular map to describe an orientable map with an orientation preserving automorphism group that is transitive on vertices and half-transitive on darts. We present a full classification of half-regular Cayley maps using the concept of skew-morphisms. We argue that half-regular Cayley maps come in two types: those that arise from two skew-morphism orbits of equal size that are both closed under inverses and those that arise from two equal-sized orbits that do not contain involutions or inverses but one contains the inverses of the other. In addition, half-regular Cayley maps of the first type are shown to be half-edge-transitive, while half-regular Cayley maps of the second type are shown to be necessarily edge-transitive. A connection between half-regular Cayley maps and regular hypermaps is also investigated. [ABSTRACT FROM AUTHOR]

A graph Γ is said to be G-arc-regular if a subgroup $G \le\operatorname{\mathsf{Aut}}(\varGamma)$ acts regularly on the arcs of Γ. In this paper connected G-arc-regular graphs are classified in the case when G contains a regular dihedral subgroup D of order 2 n whose cyclic subgroup C≤ D of index 2 is core-free in G. As an application, all regular Cayley maps over dihedral groups D, n odd, are classified. [ABSTRACT FROM AUTHOR]

Abstract: A Cayley map is -balanced if for all . Recently, Conder et al. classified the regular anti-balanced Cayley maps for abelian groups and Kwak et al. classified the regular -balanced Cayley maps for dihedral groups [8] and dicyclic groups [9]. Oh [11] classified the regular -balanced Cayley maps for semi-dihedral groups. In this paper, we classify the regular -balanced Cayley maps for cyclic groups for any . [Copyright &y& Elsevier]

Abstract: We revisit and generalize a recursive construction due to Sachs involving two graphs which increases the girth of one graph and the degree of the other. We investigate the properties of the resulting graphs in the context of cages and construct families of small graphs using geometric graphs, Paley graphs, and techniques from the theory of Cayley maps. [Copyright &y& Elsevier]

A 2-cell embedding f: X → S of a graph X into a closed orientable surface S can be described combinatorially by a pair $$\mathcal{M} = (X;\rho )$$ called a map, where ρ is a product of disjoint cycle permutations each of which is the permutation of the arc set of X initiated at the same vertex following the orientation of S. It is well known that the automorphism group of $$\mathcal{M}$$ acts semi-regularly on the arc set of X and if the action is regular then the map $$\mathcal{M}$$ and the embedding f are called regular. Let p and q be primes. Du et al. [ J. Algebraic Combin., 19, 123-141 (2004)]_classified the regular maps of graphs of order pq. In this paper all pairwise non-isomorphic regular maps of graphs of order 4 p are constructed explicitly and the genera of such regular maps are computed. As a result, there are twelve sporadic and six infinite families of regular maps of graphs of order 4 p; two of the infinite families are regular maps with the complete bipartite graphs K as underlying graphs and the other four infinite families are regular balanced Cayley maps on the groups ℤ, ℤ × ℤ and D. [ABSTRACT FROM AUTHOR]

Abstract: A cellular embedding of a connected graph is said to be Hamiltonian if every face of the embedding is bordered by a Hamiltonian cycle (a cycle containing all the vertices of ) and it is an -gonal embedding if every face of the embedding has the same length . In this paper, we establish a theory of generalized Cayley maps, including a new extension of voltage graph techniques, to show that for each even there exists a Hamiltonian embedding of such that the embedding is a Cayley map and that there is no -gonal Cayley map of if is a prime. In addition, we show that there is no Hamiltonian Cayley map of if , an odd prime and . [Copyright &y& Elsevier]

Abstract: The inherent high symmetry of Cayley maps makes them an excellent source of orientably regular maps, and the regularity of a Cayley map has been shown to be equivalent to the existence of a skew-morphism of its underlying group that has a generating orbit closed under inverses. We set to investigate the properties of the so-called -balanced skew-morphisms of abelian groups with the aim of providing the basis for a complete classification of -balanced regular Cayley maps of abelian groups. In the case of cyclic groups, we show that the only -balanced regular Cayley maps for the groups , and , an odd prime, , are the well understood balanced and antibalanced Cayley maps. [Copyright &y& Elsevier]

We consider Cayley graphs Γ over dihedral groups, dihedrants for short, which admit an automorphism group G acting regularly on the arc set of Γ. We prove that, if D≤ G≤ Aut( Γ) is a regular dihedral subgroup of G of order 2 n such that any of its index 2 cyclic subgroups is core-free in G, then Γ belongs to the family of graphs of the form $(K_{n_{1}}\otimes\cdots\otimes K_{n_{\ell}})[K_{m}^{\mathrm{c}}]$, where 2 n= n⋅⋅⋅ n m, and the numbers n are pairwise coprime. Applications to 1-regular dihedrants and Cayley maps on dihedral groups are also given. [ABSTRACT FROM AUTHOR]

In this work various maps between the space of twists and the space of finite screws are studied. Dual quaternions can be used to represent rigid-body motions, both finite screw motions and infinitesimal motions, called twists. The finite screws are elements of the group of rigid-body motions while the twists are elements of the Lie algebra of this group. The group of rigid-body displacements are represented by dual quaternions satisfying a simple relation in the algebra. The space of group elements can be though of as a six-dimensional quadric in seven-dimensional projective space, this quadric is known as the Study quadric. The twists are represented by pure dual quaternions which satisfy a degree 4 polynomial relation. This means that analytic maps between the Lie algebra and its Lie group can be written as a cubic polynomials. In order to find these polynomials a system of mutually annihilating idempotents and nilpotents is introduced. This system also helps find relations for the inverse maps. The geometry of these maps is also briefly studied. In particular, the image of a line of twists through the origin (a screw) is found. These turn out to be various rational curves in the Study quadric, a conic, twisted cubic and rational quartic for the maps under consideration. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, properties of reflexible Cayley maps for abelian groups are investigated, and as a result, it is shown that a regular Cayley map of valency greater than 2 for a cyclic group is reflexible if and only if it is anti-balanced. [Copyright &y& Elsevier]

Abstract: It is shown that every connected vertex and edge transitive graph has a normal multicover that is a connected normal edge transitive Cayley graph. Moreover, every chiral or regular map has a normal cover that is a balanced chiral or regular Cayley map, respectively. As an application, a new family of half-transitive graphs is constructed as 2-fold covers of a family of 2-arc transitive graphs admitting Suzuki groups. [Copyright &y& Elsevier]

Abstract: Given a cyclic d-tuple of integers at least 3, we consider the class of all 1-ended 3-connected d-valent planar maps such that every vertex manifests this d-tuple as the (clockwise or counterclockwise) cyclic order of covalences of its incident faces. We obtain necessary and/or sufficient conditions for the class to contain a Cayley map, a non-Cayley map whose underlying graph is a Cayley graph, a vertex-transitive graph whose subgroup of orientation-preserving automorphisms acts (or fails to act) vertex-transitively, a non-vertex-transitive map, or no planar map at all. [Copyright &y& Elsevier]

Abstract: A graph is called Frobenius if it is a connected orbital regular graph of a Frobenius group. A Frobenius map is a regular Cayley map whose underlying graph is Frobenius. In this paper, we show that almost all low-rank Frobenius graphs admit regular embeddings and enumerate non-isomorphic Frobenius maps for a given Frobenius graph. For some Frobenius groups, we classify all Frobenius maps derived from these groups. As a result, we construct some Frobenius maps with trivial exponent groups as a partial answer of a question raised by Nedela and Škoviera (Exponents of orientable maps, Proc. London Math. Soc. 75(3) (1997) 1–31). [Copyright &y& Elsevier]

A Cayley map is a Cayley graph embedded in an orientable surface such that the local rotations at every vertex are identical. In this paper, balanced regular Cayley maps for cyclic groups, dihedral groups, and generalized quaternion groups are classified. [ABSTRACT FROM AUTHOR]

The automorphism groups Aut( C( G, X)) and Aut( CM( G, X, p)) of a Cayley graph C( G, X) and a Cayley map CM( G, X, p) both contain an isomorphic copy of the underlying group G acting via left translations. In our paper, we show that both automorphism groups are rotary extensions of the group G by the stabilizer subgroup of the vertex 1 G. We use this description to derive necessary and sufficient conditions to be satisfied by a finite group in order to be the (full) automorphism group of a Cayley graph or map and classify all the finite groups that can be represented as the (full) automorphism group of some Cayley graph or map. [ABSTRACT FROM AUTHOR]