Doubly Transitive Automorphism Groups of Combinatorial Surfaces

   Abstract. The combinatorial surfaces with doubly transitive automorphism groups are classified. This is established by classifying the automorphism groups of combinatorial surfaces which act doubly transitively on the vertices of the surface. The doubly transitive automorphism groups of combinatorial surfaces are the symmetric group S ₄ , the alternating group A ₅ and the Frobenius group C ₇ · C ₆ . In each case the combinatorial surface is uniquely determined. The symmetric group S ₄ acts doubly transitively on the tetrahedron surface, the alternating group A ₅ on the triangulation of the projective plane with six vertices and the Frobenius group C ₇ · C ₆ on the Moebius torus with seven vertices.     

A design and a geometry for the group Fi 22

The Fischer group Fi ₂₂ acts as a rank 3 group of automorphisms of a symmetric 2-(14080,1444,148) design. This design does not have a doubly transitive automorphism group, since there is a partial linear space with lines of size 4 defined combinatorially from the design and preserved by its automorphism group. We investigate this geometry and determine the structure of various subspaces of it.      

Cubic vertex‐transitive graphs of order 2pq

A graph is vertex?transitive or symmetric if its automorphism group acts transitively on vertices or ordered adjacent pairs of vertices of the graph, respectively. Let G be a finite group and S a subset of G such that 1?S and S={s^?1 | s∈S}. The Cayleygraph Cay(G, S) on G with respect to S is defined as the graph with vertex set G and edge set {{g, sg} | g∈G, s∈S}. Feng and Kwak [J Combin Theory B 97 (2007), 627–646; J Austral Math Soc 81 (2006), 153–164] classified all cubic symmetric graphs of order 4p or 2p² and in this article we classify all cubic symmetric graphs of order 2pq, where p and q are distinct odd primes. Furthermore, a classification of all cubic vertex‐transitive non‐Cayley graphs of order 2pq, which were investigated extensively in the literature, is given. As a result, among others, a classification of cubic vertex‐transitive graphs of order 2pq can be deduced. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 285–302, 2010     

Neighborly combinatorial 3-manifolds with dihedral automorphism group

It is well known that for anyn≧5 the boundary complex of the cyclic 4-polytopeC(n, 4) is a neighborly combinatorial 3-sphere admitting a vertex transitive action of the dihedral groupD _n of order 2n. In this paper we present a similar series of neighborly combinatorial 3-manifolds withn≧9 vertices, each homeomorphic to the “3-dimensional Klein bottle”. Forn=9 andn=10 these examples have been observed. before by A. Altshuler and L. Steinberg. Moreover we give a computer-aided enumeration of all neighborly combinatorial 3-manifolds with such a symmetry and with at most 19 vertices. It turns out that there are only four other types, one with 10, 15, 17, 19 vertices. We also discuss the more general case of manifolds with cyclic automorphism groupC _n.     

Mendelsohn designs admitting the affine group

All Mendelsohn designs containing a Frobenius group with cyclic complement of orderv – 1 as a subgroup of the automorphism are found. Furthermore, the automorphism group of each of the designs is constructed. These designs generalize Mendelsohn's construction of Mendelsohn designs containing a certain doubly transitive automorphism group.The research on this paper was partially supported by North Texas State Faculty Research Grant #35524.     

Doubly transitive 2‐factorizations

Let be a 2‐factorization of the complete graph K_v admitting an automorphism group G acting doubly transitively on the set of vertices. The vertex‐set V(K_v) can then be identified with the point‐set of AG(n, p) and each 2‐factor of is the union of p‐cycles which are obtained from a parallel class of lines of AG(n, p) in a suitable manner, the group G being a subgroup of A G L(n, p) in this case. The proof relies on the classification of 2‐(v, k, 1) designs admitting a doubly transitive automorphism group. The same conclusion holds even if G is only assumed to act doubly homogeneously. © 2006 Wiley Periodicals, Inc. J Combin Designs     

A new 5-arc-transitive cubic graph

A connected cubic graph having 75,600 vertices is shown to exist, with the symmetric group S₁₀ as a group of automorphisms acting transitively on its 5-arcs. This graph is not bipartite, nor is it a covering of any other known 4- or 5-arc-transitive graph.     

Tetravalent vertex‐transitive graphs of order 4p

A graph is vertex‐transitive if its automorphism group acts transitively on vertices of the graph. A vertex‐transitive graph is a Cayley graph if its automorphism group contains a subgroup acting regularly on its vertices. In this article, the tetravalent vertex‐transitive non‐Cayley graphs of order 4p are classified for each prime p. As a result, there are one sporadic and five infinite families of such graphs, of which the sporadic one has order 20, and one infinite family exists for every prime p>3, two families exist if and only if p≡1 (mod 8) and the other two families exist if and only if p≡1 (mod 4). For each family there is a unique graph for a given order. © 2011 Wiley Periodicals, Inc.     

Realizing finite edge‐transitive orientable maps

J.E. Graver and M.E. Watkins, Memoirs Am. Math. Soc. 126 (601) ( 5 ) established that the automorphism group of an edge‐transitive, locally finite map manifests one of exactly 14 algebraically consistent combinations (called types) of the kinds of stabilizers of its edges, its vertices, its faces, and its Petrie walks. Exactly eight of these types are realized by infinite, locally finite maps in the plane. H.S.M. Coxeter (Regular Polytopes, 2nd ed., McMillan, New York, 1963) had previously observed that the nine finite edge‐transitive planar maps realize three of the eight planar types. In the present work, we show that for each of the 14 types and each integer n ≥ 11 such that n ≡ 3,11 (mod 12), there exist finite, orientable, edge‐transitive maps whose various stabilizers conform to the given type and whose automorphism groups are (abstractly) isomorphic to the symmetric group Sym(n). Exactly seven of these types (not a subset of the planar eight) are shown to admit infinite families of finite, edge‐transitive maps on the torus, and their automorphism groups are determined explicitly. Thus all finite, edge‐transitive toroidal maps are classified according to this schema. Finally, it is shown that exactly one of the 14 types can be realized as an abelian group of an edge‐transitive map, namely, as ?_n × ?₂ where n ≡ 2 (mod 4). © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 1–34, 2001     

Sharply transitive 1-factorizations of the complete graph with an invariant 1-factor

We give some simple characterizations of those n for which K_n has a sharply transitive 1-factorization with an assigned automorphism group that acts sharply transitively on the vertex set and also fixes a 1-factor. © 1994 John Wiley & Sons, Inc.     

New symmetric designs and nonabelian difference sets with parameters(100, 45, 20)

Six nonisomorphic new symmetric designs with parameters (100, 45, 20) are constructed by action of the Frobenius group E₂₅ · Z₁₂. This group proves to be their full automorphism group. Its Frobenius subgroup of order 100 acts on the designs as their nonabelian Singer group. The result is presented through six nonisomorphic new nonabelian (100, 45, 20) difference sets as well. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 291–299, 2000     

ON ARC-TRANSITIVE CIRCULANT DIGRAPHS

Denote by c,(s)the circulant digraph with vertex set zn=[0,1,2……n-1]and symbol set s(≠-s)∈zn\[0].let x be the automorphism group of cn(S)and xo the stabilizer of o in x.then cn(S)is arctransitive if and only if xo acts transitively on s.in this paper,co(S)with xo is being the symmetric group is characterized by its symbot set .by the way all the arctransitive clcculant digraphs of degree 2are given.     

Unitary graphs and classification of a family of symmetric graphs with complete quotients

A finite graph Γ is called G-symmetric if G is a group of automorphisms of Γ which is transitive on the set of ordered pairs of adjacent vertices of Γ. We study a family of symmetric graphs, called the unitary graphs, whose vertices are flags of the Hermitian unital and whose adjacency relations are determined by certain elements of the underlying finite fields. Such graphs admit the unitary groups as groups of automorphisms, and play a significant role in the classification of a family of symmetric graphs with complete quotients such that an associated incidence structure is a doubly point-transitive linear space. We give this classification in the paper and also investigate combinatorial properties of the unitary graphs.     

Alternating groups and flag‐transitive 2‐(v,k, 4) symmetric designs

In this article, we study the classification of flag‐transitive, point‐primitive 2‐ (v, k, 4) symmetric designs. We prove that if the socle of the automorphism group G of a flag‐transitive, point‐primitive nontrivial 2‐ (v, k, 4) symmetric design ?? is an alternating group A_n for n≥5, then (v, k) = (15, 8) and ?? is one of the following: (i) The points of ?? are those of the projective space PG(3, 2) and the blocks are the complements of the planes of PG(3, 2), G = A₇ or A₈, and the stabilizer G_x of a point x of ?? is L₃(2) or AGL₃(2), respectively. (ii) The points of ?? are the edges of the complete graph K₆ and the blocks are the complete bipartite subgraphs K_{2, 4} of K₆, G = A₆ or S₆, and G_x = S₄ or S₄ × Z₂, respectively. © 2011 Wiley Periodicals, Inc. J Combin Designs 19:475‐483, 2011     

Finite-dimensional pointed Hopf algebras with alternating groups are trivial

It is shown that Nichols algebras over alternating groups \mathbb A_m{\mathbb A_m} (m ≥ 5) are infinite dimensional. This proves that any complex finite dimensional pointed Hopf algebra with group of group-likes isomorphic to \mathbb A_m{\mathbb A_m} is isomorphic to the group algebra. In a similar fashion, it is shown that the Nichols algebras over the symmetric groups \mathbb S_m{\mathbb S_m} are all infinite-dimensional, except maybe those related to the transpositions considered in Fomin and Kirillov (Progr Math 172:146–182, 1999), and the class of type (2, 3) in \mathbb S₅{\mathbb S_5}. We also show that any simple rack X arising from a symmetric group, with the exception of a small list, collapse, in the sense that the Nichols algebra \mathfrak B(X, q){\mathfrak B(X, \bf q)} is infinite dimensional, q an arbitrary cocycle.     

Tetravalent half‐edge‐transitive graphs and non‐normal Cayley graphs

Let X be a vertex‐transitive graph, that is, the automorphism group Aut(X) of X is transitive on the vertex set of X. The graph X is said to be symmetric if Aut(X) is transitive on the arc set of X. suppose that Aut(X) has two orbits of the same length on the arc set of X. Then X is said to be half‐arc‐transitive or half‐edge‐transitive if Aut(X) has one or two orbits on the edge set of X, respectively. Stabilizers of symmetric and half‐arc‐transitive graphs have been investigated by many authors. For example, see Tutte [Canad J Math 11 (1959), 621–624] and Conder and Maru?i? [J Combin Theory Ser B 88 (2003), 67–76]. It is trivial to construct connected tetravalent symmetric graphs with arbitrarily large stabilizers, and by Maru?i? [Discrete Math 299 (2005), 180–193], connected tetravalent half‐arc‐transitive graphs can have arbitrarily large stabilizers. In this article, we show that connected tetravalent half‐edge‐transitive graphs can also have arbitrarily large stabilizers. A Cayley graph Cay(G, S) on a group G is said to be normal if the right regular representation R(G) of G is normal in Aut(Cay(G, S)). There are only a few known examples of connected tetravalent non‐normal Cayley graphs on non‐abelian simple groups. In this article, we give a sufficient condition for non‐normal Cayley graphs and by using the condition, infinitely many connected tetravalent non‐normal Cayley graphs are constructed. As an application, all connected tetravalent non‐normal Cayley graphs on the alternating group A₆ are determined. © 2011 Wiley Periodicals, Inc. J Graph Theory     

Generalized quadrangles admitting a sharply transitive Heisenberg group

All known finite generalized quadrangles that admit an automorphism group acting sharply transitively on their point set arise by Payne derivation from thick elation generalized quadrangles of order s with a regular point. In these examples only two groups occur: elementary abelian groups of even order and odd order Heisenberg groups of dimension 3. In [2] the authors determined all generalized quadrangles admitting an abelian group with a sharply transitive point action. Here, we classify thick finite generalized quadrangles admitting an odd order Heisenberg group of dimension 3 acting sharply transitively on the points. In fact our more general result comes close to a complete solution of classifying odd order Singer p-groups.      

Constructing even radius tightly attached half-arc-transitive graphs of valency four

A finite graph X is half-arc-transitive if its automorphism group is transitive on vertices and edges, but not on arcs. When X is tetravalent, the automorphism group induces an orientation on the edges and a cycle of X is called an alternating cycle if its consecutive edges in the cycle have opposite orientations. All alternating cycles of X have the same length and half of this length is called the radius of X. The graph X is said to be tightly attached if any two adjacent alternating cycles intersect in the same number of vertices equal to the radius of X. Marušič (J. Comb. Theory B, 73, 41–76, 1998) classified odd radius tightly attached tetravalent half-arc-transitive graphs. In this paper, we classify the half-arc-transitive regular coverings of the complete bipartite graph K _4,4 whose covering transformation group is cyclic of prime-power order and whose fibre-preserving group contains a half-arc-transitive subgroup. As a result, two new infinite families of even radius tightly attached tetravalent half-arc-transitive graphs are constructed, introducing the first infinite families of tetravalent half-arc-transitive graphs of 2-power orders.      

A combinatorial approach to transitive extensions of generously unitransitive permutation groups

Motivated by symmetric association schemes (which are known to approximate generously unitransitive group actions), we formulate combinatorial approximations to transitive extensions of generously unitransitive permutation groups. Specifically, the notions of compatible and coherent partitions are suggested and investigated in terms of the orbits of an ambient group (H, Ω) on the k‐subsets of Ω, k=2, 3, 4. We apply these ideas to investigate transitive extensions of the automorphism groups of the classical Johnson and Hamming schemes. In the latter case, we further provide algorithmic details and computer‐generated data for the particular series of Hamming schemes H(m, 3), m⩾2. Finally, our approach is compared to the concept of a symmetric association scheme on triples in the sense of Mesner and Bhattacharya. © 2010 Wiley Periodicals, Inc. J Combin Designs 18:369–391, 2010     

ON A SELF DUAL 3-SPHERE OF PETER MCMULLEN

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. This revised version was published online in August 2006 with corrections to the Cover Date.