Group Of Orientation-preserving Automorphisms Research Articles

Hole operations on Hurwitz maps

For a given group G the orientably regular maps with orientation-preserving automorphism group G are used as the vertices of a graph O(G), with undirected and directed edges showing the effect of duality and hole operations on these maps. Some examples of these graphs are given, including several for small Hurwitz groups. For some G, such as the affine groups AGL1(2e), the graph O(G) is connected, whereas for some other infinite families, such as the alternating and symmetric groups, the number of connected components is unbounded.

On Nilpotent Orientably-Regular Maps of Nilpotency Class $4$

By a nilpotent map we mean an orientably regular map whose orientation preserving automorphism group is nilpotent. The nilpotent maps are concluded to the maps whose automorphism group is a $2$-group and a complete classification of nilpotent maps of (nilpotency) class $2$ is given by Malnič et al. in [European J. Combin. 33 (2012), 1974-1986]. It is proved by Conder et al. in [J. Algebraic Combin. 44 (2016), 863-874] that given the class, there are finitely many simple nilpotent maps. However, for the nilpotent maps with multiple edges and given class, since its automorphism group may be infinitely big, it is impossible to list it by a computer. Therefore, to classify the nilpotent maps with small class $c$ is necessary and interesting. In this paper, the nilpotent maps of class $4$ will be determined.

A classification of primer hypermaps with a product of two primes number of hyperfaces

A hypermap is a cellular embedding of a connected bipartite graph G into a compact and connected surface S without border. The vertices of G lie in distinct partitions, colored black and white, and are called respectively the hypervertices and hyperedges of the hypermap, while the connected regions of G∖S are the hyperfaces. A hypermap H is called orientable if the underlying surface S is orientable. An orientable hypermap is called regular if its orientation preserving automorphism group G acts regularly on the flags (hypervertex, hyperedge and hyperface incident triples), and further, it is called (face-)primer if G induces a faithful action on their hyperfaces. In Breda d’Azevedo and Fernandes (2011), a classification of the primer hypermap with a prime number of hyperfaces is given. In this paper, a classification will be given, for all primer hypermaps with a product of two primes hyperfaces, see Theorem 5.1.

Non-abelian almost totally branched coverings over the platonic maps

A map is a 2-cell embedding of a connected graph into a closed surface. A map is orientable if the supporting surface is orientable. An orientable map is regular if its group of orientation-preserving automorphisms acts transitively on the darts. Using an equivalent algebraic description of regular maps and their coverings, we employ the theory of group extensions to classify the almost totally branched coverings of the platonic maps with non-abelian covering transformation groups, generalising the results of Hu, Nedela and Wang.

Symmetry Type Graphs of Polytopes and Maniplexes

We extend the notion of symmetry type graphs of maps to include maniplexes and (abstract) polytopes, using them to study k-orbit maniplexes (where the automorphism group has k orbits on flags). In particular, we show that there are no fully-transitive k-orbit 3-maniplexes with k > 1 an odd number. We classify 3-orbit maniplexes and determine all face transitivities for 3- and 4-orbit maniplexes. Moreover, we give generators of the automorphism group of a maniplex, given its symmetry type graph. Finally, we extend these notions to oriented maniplexes, and we provide a classification of oriented 2-orbit maniplexes and a generating set for their orientation-preserving automorphism group.

Half-Regular Cayley Maps

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.

Distinguishing Maps II: General Case

A group $A$ acting faithfully on a set $X$ has distinguishing number $k$, written $D(A,X)=k$, if there is a coloring of the elements of $X$ with $k$ colors such that no nonidentity element of $A$ is color-preserving, and no such coloring with fewer than $k$ colors exists. Given a map $M$ with vertex set $V$ and automorphism group $Aut(M)$, let $D(M)=D(Aut(M),V)$. If $M$ is orientable, let $D^+(M)=D(Aut^+(M),V)$, where $Aut^+(M)$ is the group of orientation-preserving automorphisms. In a previous paper, the author showed there are four maps $M$ with $D^+(M)>2$. In this paper, a complete classification is given for the graphs underlying maps with $D(M)>2$. There are $31$ such graphs, $22$ having no vertices of valence $1$ or $2$, and all have at most $10$ vertices.

On orientation-preserving automorphisms of Hamiltonian cycles in the N-dimensional Boolean cube

We obtain an upper bound for the order of the group of orientation-preserving automorphisms of a Hamiltonian cycle in the Boolean n-cube. We prove that the existence of a Hamiltonian cycle with the order of the group of orientation-preserving automorphisms attaining this upper bound is equivalent to the existence of a Hamiltonian cycle with an additional condition on the graph of orbits of a fixed automorphism of the n-cube.

The strong symmetric genus and generalized symmetric groups G(n, 3)

The generalized symmetric groups are defined to be the groups G(n, m) = ℤm ≀ Σm where n, m ∈ ℤ+. The strong symmetric genus of a finite group G is the smallest genus of a closed orientable topological surface on which G acts faithfully as a group of orientation-preserving automorphisms. The present paper extends work on the strong symmetric genus by Conder, who studied the symmetric groups, which are the groups G(n, 1), and the author, who studied the hyperoctahedral groups, which are the groups G(n, 2). We determine the strong symmetric genus of the groups G(n, 3).

Regular Cayley maps for finite abelian groups

A regular Cayley map for a finite group A is an orientable map whose orientation-preserving automorphism group G acts regularly on the directed edge set and has a subgroup isomorphic to A that acts regularly on the vertex set. This paper considers the problem of determining which abelian groups have regular Cayley maps. The analysis is purely algebraic, involving the structure of the canonical form for A. The case when A is normal in G involves the relationship between the rank of A and the exponent of the automorphism group of A, and the general case uses Ito's theorem to analyze the factorization G = AY, where Y is the (cyclic) stabilizer of a vertex.

Half-arc-transitive graphs and chiral hypermaps

A subgroup G of automorphisms of a graph X is said to be 1 2 - arc- transitive if it is vertex- and edge- but not arc-transitive. The graph X is said to be 1 2 - arc- transitive if Aut X is 1 2 -arc-transitive. The interplay of two different concepts, maps and hypermaps on one side and 1 2 -arc-transitive group actions on graphs on the other, is investigated. The correspondence between regular maps and 1 2 -arc-transitive group actions on graphs of valency 4 given via the well known concept of medial graphs (European J. Combin. 19 (1998) 345) is generalised. Any orientably regular hypermap H gives rise to a uniquely determined medial map whose underlying graph Y admits a 1 2 -arc-transitive group action of the automorphism group G of the original hypermap H . Moreover, the vertex stabiliser of the action of G on Y is cyclic. On the other hand, given graph X and G≤ Aut X acting 1 2 -arc-transitively with a cyclic vertex stabiliser, we can construct an orientably regular hypermap H with G being the orientation preserving automorphism group. In particularly, if the graph X is 1 2 -arc-transitive, the corresponding hypermap is necessarily chiral, that is, not isomorphic to its mirror image. Note that the associated 1 2 -arc-transitive group action on the medial graph induced by a map always has a stabiliser of order two, while when it is induced by a (pure) hypermap the stabiliser can be cyclic of arbitrarily large order. Hence moving from maps to hypermaps increases our chance of getting different types of 1 2 -arc-transitive group action. Indeed, in last section we have applied general results to construct 1 2 -arc-transitive graphs with cycle stabilisers of arbitrarily large orders.