Cayley Maps Research Articles

On exact products of a cyclic group and a dihedral group

A finite group G is an exact product of two subgroups A and B if G = AB and A ∩ B = { 1 G } . 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 D 2 k . As an application the regular generalized Cayley maps of odd valency over cyclic groups are classified.

Generalized Cayley maps and their Petrie duals

Generalized Cayley maps and their Petrie duals

Almost all Cayley maps are mapical regular representations

Cayley maps are combinatorial structures built upon Cayley graphs on a group. As such the original group embeds in their group of automorphisms, and one can ask in which situation the two coincide (one then calls the Cayley map  a mapical regular representation or MRR) and with what probability. The first question was answered by Jajcay. In this paper we tackle the probabilistic version, and prove that as groups get larger the proportion of MRRs among all Cayley Maps approaches 1.

Regular Cayley maps of elementary abelian p-groups: Classification and enumeration

Recently, regular Cayley maps of cyclic groups and dihedral groups have been classified in [7] and [20], respectively. A nature question is to classify regular Cayley maps of elementary abelian p-groups Zpn. In this paper, a complete classification of regular Cayley maps of Zpn is given and moreover, the number of these maps and their genera are enumerated.

Skew product groups for monolithic groups

Skew morphisms, which generalise automorphisms for groups, provide a fundamental tool for the study of regular Cayley maps and, more generally, for finite groups with a complementary factorisation $G=BY$, where $Y$ is cyclic and core-free in $G$. In this paper, we classify all examples in which $B$ is monolithic (meaning that it has a unique minimal normal subgroup, and that subgroup is not abelian) and core-free in $G$. As a consequence, we obtain a classification of all proper skew morphisms of finite non-abelian simple groups.

Skew-morphisms of nonabelian characteristically simple groups

A skew-morphism of a finite group G is a permutation σ on G fixing the identity element such that the product of 〈σ〉 with the left regular representation of G forms a permutation group on G. This permutation group is called the skew-product group of σ. The skew-morphism was introduced as an algebraic tool to investigate regular Cayley maps. In this paper, we characterize skew-products of skew-morphisms of finite nonabelian characteristically simple groups (see Theorem 1.2) and the corresponding regular Cayley maps (see Theorem 1.6).

Review of the exponential and Cayley map on SE(3) as relevant for Lie group integration of the generalized Poisson equation and flexible multibody systems

The exponential and Cayley maps on SE(3) are the prevailing coordinate maps used in Lie group integration schemes for rigid body and flexible body systems. Such geometric integrators are the Munthe–Kaas and generalized- α schemes, which involve the differential and its directional derivative of the respective coordinate map. Relevant closed form expressions, which were reported over the last two decades, are scattered in the literature, and some are reported without proof. This paper provides a reference summarizing all relevant closed-form relations along with the relevant proofs, including the right-trivialized differential of the exponential and Cayley map and their directional derivatives (resembling the Hessian). The latter gives rise to an implicit generalized- α scheme for rigid/flexible multibody systems in terms of the Cayley map with improved computational efficiency.

Regular Cayley maps for dihedral groups

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 Mi(n,r), 1≤i≤5, are derived, where 2n is the order of the regular dihedral group and r is an integer satisfying certain arithmetical conditions. For each map Mi(n,r), we determine its valence and covalence, and also describe the structure of the group Aut∘Mi(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.

Frobenius groups which are the automorphism groups of orientably-regular maps

In this paper, we prove that a Frobenius group (except for those which are dihedral groups) can only be the automorphism group of an orientably-regular chiral map. The necessary and sufficient conditions for a Frobenius group to be the automorphism group of an orientably-regular chiral map are given. Furthermore, these orientably-regular chiral maps with Frobenius automorphisms are proved to be normal Cayley maps. Frobenius groups conforming to these conditions are also constructed.

Regular balanced Cayley maps on nonabelian metacyclic groups of odd order

In this paper, we show that nonabelian metacyclic groups of odd order do not have regular balanced Cayley maps.

Kinematic Comparison of Exponential and Cayley Maps in the Planar Motion Group

In this work, the exponential and the Cayley maps, from the Lie algebra $\mathfrak{se(2)}$ of the planar motion group $SE(2)$, to the group itself are studied. The comparison between these maps on $SE(2)$ is given by using the Rodrigues vector. A three joint mechanism is discussed as an application.

Motor Parameterization

In this paper, we consider several parameterizations of rigid transformations using motors in 3-D conformal geometric algebra. In particular, we present parameterizations based on the exponential, outer exponential, and Cayley maps of bivectors, as well as a map based on a first-order approximation of the exponential followed by orthogonal projection onto the group manifold. We relate these parameterizations to the matrix representations of rigid transformations in the 3-D special Euclidean group. Moreover, we present how these maps can be used to form retraction maps for use in manifold optimization; retractions being approximations of the exponential map that preserve the convergence properties of the optimization method while being less computationally expensive, and, for the presented maps, also easier to implement.

The Cayley Isomorphism Property for Cayley Maps

The Cayley Isomorphism property for combinatorial objects was introduced by L. Babai in 1977. Since then it has been intensively studied for binary relational structures: graphs, digraphs, colored graphs etc. In this paper we study this property for oriented Cayley maps. A Cayley map is a Cayley graph provided by a cyclic rotation of its connection set. If the underlying graph is connected, then the map is an embedding of a Cayley graph into an oriented surface with the same cyclic rotation around every vertex.Two Cayley maps are called Cayley isomorphic if there exists a map isomorphism between them which is a group isomorphism too. We say that a finite group $H$ is a CIM-group if any two Cayley maps over $H$ are isomorphic if and only if they are Cayley isomorphic.The paper contains two main results regarding CIM-groups. The first one provides necessary conditons for being a CIM-group. It shows that a CIM-group should be one of the following$$\mathbb{Z}_m\times\mathbb{Z}_2^r, \\mathbb{Z}_m\times\mathbb{Z}_{4},\\mathbb{Z}_m\times\mathbb{Z}_{8}, \ \mathbb{Z}_m\times Q_8, \\mathbb{Z}_m\rtimes\mathbb{Z}_{2^e}, e=1,2,3,$$ where $m$ is an odd square-free number and $r$ a non-negative integer. Our second main result shows that the groups $\mathbb{Z}_m\times\mathbb{Z}_2^r$, $\mathbb{Z}_m\times\mathbb{Z}_{4}$, $\mathbb{Z}_m\times Q_8$ contained in the above list are indeed CIM-groups.

Classification of regular balanced Cayley maps of minimal non-abelian metacyclic groups

In this paper, we classify the regular balanced Cayley maps of minimal non-abelian metacyclic groups. Besides the quaternion group Q8, there are two infinite families of such groups which are denoted by Mp, q(m, r) and Mp(n, m), respectively. Firstly, we prove that there are regular balanced Cayley maps of Mp, q(m, r) if and only if q = 2 and we list all of them up to isomorphism. Secondly, we prove that there are regular balanced Cayley maps of Mp(n, m) if and only if p = 2 and n = m or n = m + 1 and there is exactly one such map up to isomorphism in either case. Finally, as a corollary, we prove that any metacyclic p-group for odd prime number p does not have regular balanced Cayley maps.

Classification of reflexible Cayley maps for dihedral groups

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.

Quotients of polynomial rings and regular [formula omitted]-balanced Cayley maps on abelian groups

Given a finite group Γ, a regular t-balanced Cayley map (RBCMt for short) is a regular Cayley map CM(G,Ω,ρ) such that ρ(ω)−1=ρt(ω) for all ω∈Ω. In this paper, we clarify a connection between quotients of polynomial rings and RBCMt’s on abelian groups, so as to propose a new approach for classifying RBCMt’s. We obtain many new results, in particular, a complete classification for RBCMt’s on abelian 2-groups.

Automorphism groups of Cayley graphs generated by block transpositions and regular Cayley maps

This paper deals with the Cayley graph Cay(Symn,Tn), 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(Symn,Tn)) is the product of the left translation group and a dihedral group Dn+1 of order 2(n+1). The proof uses several properties of the subgraph Γ of Cay(Symn,Tn) induced by the set Tn. In particular, Γ is a 2(n−2)-regular graph whose automorphism group is Dn+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 Dn+1 of order n+1 with regular Cayley maps on Symn 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 Symn.

On the skew-morphisms of dihedral groups

Abstract A skew-morphism φ of a finite group G is a permutation on G such that φ ⁢ ( 1 ) = 1 ${\varphi(1)\hskip-0.853583pt=\hskip-0.853583pt1}$ and φ ⁢ ( g ⁢ h ) = φ ⁢ ( g ) ⁢ φ π ⁢ ( g ) ⁢ ( h ) ${\varphi(gh)=\varphi(g)\varphi^{\pi(g)}(h)}$ for all g , h ∈ G ${g,h\in G}$ , where π is a function from G to 𝐙 | φ | ${\mathbf{Z}_{|\varphi|}}$ , called the power function of φ. Furthermore, we say φ is of skew-type k, provided π takes on exactly k values in 𝐙 | φ | ${\mathbf{Z}_{|\varphi|}}$ , and call the set Ker ⁡ φ = { g ∈ G ∣ π ⁢ ( g ) = 1 } ${\operatorname{Ker}\varphi=\{g\in G\mid\pi(g)=1\}}$ the kernel of φ, which is actually a subgroup of G (see [6]). Though skew-morphism is a pure group theoretical concept, it is closely related to regular Cayley maps. We have a long term goal to classify all regular Cayley maps for dihedral groups. In this paper, we mainly work on the skew-morphisms of dihedral groups. Let φ be a skew-morphism of the dihedral groups D 2 ⁢ n = 〈 a , b ∣ a n = b 2 = ( a ⁢ b ) 2 = 1 〉 ${D_{2n}=\langle a,b\mid a^{n}=b^{2}=(ab)^{2}=1\rangle}$ . We prove that the kernel of φ is contained in the cyclic subgroup 〈 a 〉 ${\langle a\rangle}$ if and only if φ is of skew-type 4 and preserves 〈 a 〉 ${\langle a\rangle}$ . In the case of φ preserving 〈 a 〉 ${\langle a\rangle}$ , we give some characterizations of φ, in particular, we prove that φ is of skew-type 1, 2 or 4. Working from these characterizations, an infinite family of skew-morphisms of D 2 ⁢ n ${D_{2n}}$ are constructed, all of which are of skew-type 4 and have kernel 〈 a 2 〉 ${\langle a^{2}\rangle}$ , which gives a positive answer to an open problem raised recently by M. Conder [1].

Regular Cayley maps on dihedral groups with the smallest kernel

Let $$\mathcal {M}={{\mathrm{CM}}}(D_n,X,p)$$M=CM(Dn,X,p) be a regular Cayley map on the dihedral group $$D_n$$Dn of order $$2n, n \ge 2,$$2n,nź2, and let $$\psi $$ź be the skew-morphism associated with $$\mathcal {M}$$M. In this paper it is shown that the kernel $${{\mathrm{Ker}}}\psi $$Kerź of the skew-morphism $$\psi $$ź is a dihedral subgroup of $$D_n$$Dn and if $$n \ne 3,$$nź3, then the kernel $${{\mathrm{Ker}}}\psi $$Kerź is of order at least 4. Moreover, all $$\mathcal {M}$$M are classified for which $${{\mathrm{Ker}}}\psi $$Kerź is of order 4. In particular, besides four sporadic maps on 4, 4, 8 and 12 vertices, respectively, two infinite families of non-t-balanced Cayley maps on $$D_n$$Dn are obtained.

Cyclic complements and skew morphisms of groups

A skew morphism of a group is a generalisation of an automorphism, arising in the context of regular Cayley maps or of groups expressible as a product AB of subgroups A and B with B cyclic and A∩B={1}. A skew morphism of a group A is a bijection φ:A→A fixing the identity element of A and having the property that φ(xy)=φ(x)φπ(x)(y) for all x,y∈A, where π(x) depends only on x. The kernel of φ is the subgroup of all x∈A for which π(x)=1.In this paper, we present a number of previously unknown properties of skew morphisms, one being that if A is any finite group, then the order of every skew morphism of A is less than |A|, and another being that the kernel of every skew morphism of a non-trivial finite group is non-trivial. We also prove a number of theorems about skew morphisms of finite abelian groups, some of which simplify or extend recent theorems of Kovács and Nedela [13]. For example, we determine all skew morphisms of the finite abelian groups whose order is prime, or the square of a prime, or the product of two distinct primes. In addition, we completely determine the finite abelian groups for which every skew morphism is an automorphism.