Orthogonal packings inPG(5, 2)

Aspread inPG(n, q) is a set of lines which partitions the point set. A packing inPG(n, q) (n odd) is a partition of the lines into spreads. Two packings ofPG(n, q) are calledorthogonal if and only if any two spreads, one from each packing, have at most one line in common. Recently, R. D. Baker has shown the existence of a pair of orthogonal packings inPG(5, 2). In this paper we enumerate all packings inPG(5, 2) having both an automorphism of order 31 and the Frobenius automorphism. We find all pairs of orthogonal packings of the above type and display a set of six mutually orthogonal packings. Previously the largest set of orthogonal packings known inPG(5, 2) was two.     

Classification of spreads ofPG(3, 4)∖PG(3, 2)

We show that the 800 spreads ofPG(3, 4)PG(3, 2) fall into three orbits of sizes 120, 120 and 560, under the action of its automorphism group.     

Automorphisms of higher‐dimensional Hadamard matrices

This article derives from first principles a definition of equivalence for higher‐dimensional Hadamard matrices and thereby a definition of the automorphism group for higher‐dimensional Hadamard matrices. Our procedure is quite general and could be applied to other kinds of designs for which there are no established definitions for equivalence or automorphism. Given a two‐dimensional Hadamard matrix H of order ν, there is a Product Construction which gives an order ν proper n‐dimensional Hadamard matrix P⁽ⁿ⁾(H). We apply our ideas to the matrices P⁽ⁿ⁾(H). We prove that there is a constant c > 1 such that any Hadamard matrix H of order ν > 2 gives rise via the Product Construction to cν inequivalent proper three‐dimensional Hadamard matrices of order ν. This corrects an erroneous assertion made in the literature that ”P⁽ⁿ⁾(H) is equivalent to “P⁽ⁿ⁾(H′) whenever H is equivalent to H′.” We also show how the automorphism group of P⁽ⁿ⁾(H) depends on the structure of the automorphism group of H. As an application of the above ideas, we determine the automorphism group of P⁽ⁿ⁾(Hk) when Hk is a Sylvester Hadamard matrix of order 2^k. For ν = 4, we exhibit three distinct families of inequivalent Product Construction matrices P⁽ⁿ⁾(H) where H is equivalent to H₂. These matrices each have large but non‐isomorphic automorphism groups. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 507–544, 2008     

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     

Nilpotent 1‐factorizations of the complete graph

For which groups G of even order 2n does a 1‐factorization of the complete graph K_2n exist with the property of admitting G as a sharply vertex‐transitive automorphism group? The complete answer is still unknown. Using the definition of a starter in G introduced in 4 , we give a positive answer for new classes of groups; for example, the nilpotent groups with either an abelian Sylow 2‐subgroup or a non‐abelian Sylow 2‐subgroup which possesses a cyclic subgroup of index 2. Further considerations are given in case the automorphism group G fixes a 1‐factor. © 2005 Wiley Periodicals, Inc. J Combin Designs     

A family of non class-regular symmetric transversal designs of spread type

Many classes of symmetric transversal designs have been constructed from generalized Hadamard matrices and they are necessarily class regular. In (Hiramine, Des Codes Cryptogr 56:21–33, 2010) we constructed symmetric transversal designs using spreads of \mathbbZ_p²ⁿ{\mathbb{Z}_p^{2n}} with p a prime. In this article we show that most of them admit no class regular automorphism groups. This implies that they are never obtained from generalized Hadamard matrices. As far as we know, this is the first infinite family of non class-regular symmetric transversal designs.     

The graphical complexity of direct products of permutation groups

In this article, we improve known results, and, with one exceptional case, prove that when k≥3, the direct product of the automorphism groups of graphs whose edges are colored using k colors, is itself the automorphism group of a graph whose edges are colored using k colors. We have handled the case k = 2 in an earlier article. We prove similar results for directed edge‐colored graphs. © 2010 Wiley Periodicals, Inc. J Graph Theory 66:303‐318, 2011     

Resolutions of PG(5, 2) with point‐cyclic automorphism group

A t‐(υ, k, λ) design is a set of υ points together with a collection of its k‐subsets called blocks so that all subsets of t points are contained in exactly λ blocks. The d‐dimensional projective geometry over GF(q), PG(d, q), is a 2‐(q^d + q^d−1 + … + q + 1, q + 1, 1) design when we take its points as the points of the design and its lines as the blocks of the design. A 2‐(υ, k, 1) design is said to be resolvable if the blocks can be partitioned as ℛ = {R₁, R₂, …, R_s}, where s = (υ − 1)/(k−1) and each R_i consists of υ/k disjoint blocks. If a resolvable design has an automorphism σ which acts as a cycle of length υ on the points and ℛ^σ = ℛ, then the design is said to be point‐cyclically resolvable. The design associated with PG(5, 2) is known to be resolvable and in this paper, it is shown to be point‐cyclically resolvable by enumerating all inequivalent resolutions which are invariant under a cyclic automorphism group G = 〈σ〉 where σ is a cycle of length 63. These resolutions are the only resolutions which admit a point‐transitive automorphism group. Furthermore, some necessary conditions for the point‐cyclic resolvability of 2‐(υ, k, 1) designs are also given. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 2–14, 2000     

Near‐automorphisms of Latin squares

We define a near‐automorphism α of a Latin square L to be an isomorphism such that L and α L differ only within a 2 × 2 subsquare. We prove that for all n≥2 except n∈{3, 4}, there exists a Latin square which exhibits a near‐automorphism. We also show that if α has the cycle structure (2, n ? 2), then L exists if and only if n≡2 (mod 4), and can be constructed from a special type of partial orthomorphism. Along the way, we generalize a theorem by Marshall Hall, which states that any Latin rectangle can be extended to a Latin square. We also show that if α has at least 2 fixed points, then L must contain two disjoint non‐trivial subsquares. Copyright © 2011 John Wiley & Sons, Ltd. 19:365‐377, 2011     

Locally symmetric left invariant Riemannian metrics on 3‐dimensional Lie groups

In this paper, we use the powerful tool Milnor bases to determine all the locally symmetric left invariant Riemannian metrics up to automorphism, on 3‐dimensional connected and simply connected Lie groups, by solving system of polynomial equations of constants structure of each Lie algebra . Moreover, we show that E ₀(2) is the only 3‐dimensional Lie group with locally symmetric left invariant Riemannian metrics which are not symmetric.     

Steiner systems with automorphism groups PSL(2, 71), PSL(2, 83), and P∑L(2, 35)

The first 5-(72, 6, 1) designs with automorphism group PSL(2, 71) were found by Mills [10]. We presently enumerate all 5-(72, 6, 1) designs with this automorphism group. There are in all 926299 non-isomorphic designs. We show that a necessary condition for semiregular5-(v, 6, 1) designs with automorphism group PSL(2, v 1) to exist is thatv=84, 228 (mod 360). In particular, there are exactly 3 non-isomorphic semiregular 5-(84, 6, 1) designs with automorphism group PSL(2, 83). There are at least 6450 non-isomorphic 5-(244, 6, 1) designs with automorphism group PL(2, 3⁵).     

On 2‐factorizations of the complete graph: From the k‐pyramidal to the universal property

We consider 2‐factorizations of complete graphs that possess an automorphism group fixing k?0 vertices and acting sharply transitively on the others. We study the structures of such factorizations and consider the cases in which the group is either abelian or dihedral in some more details. Combining results of the first part of the paper with a result of D. Bryant, J Combin Des, 12 (2004), 147–155, we prove that the class of 2‐factorizations of complete graphs is universal. Namely each finite group is the full automorphism group of a 2‐factorization of the class. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 211‐228, 2009     

One‐factorizations of complete graphs with vertex‐regular automorphism groups

We consider one‐factorizations of K_2n possessing an automorphism group acting regularly (sharply transitively) on vertices. We present some upper bounds on the number of one‐factors which are fixed by the group; further information is obtained when equality holds in these bounds. The case where the group is dihedral is studied in some detail, with some non‐existence statements in case the number of fixed one‐factors is as large as possible. Constructions both for dihedral groups and for some classes of abelian groups are given. © 2002 John Wiley & Sons, Inc. J Combin Designs 10: 1–16, 2002     

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     

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.     

Regular representations of finite groups as automorphism groups of k‐tournaments

It was shown by Babai and Imrich [2] that every finite group of odd order except and admits a regular representation as the automorphism group of a tournament. Here, we show that for k ≥ 3, every finite group whose order is relatively prime to and strictly larger than k admits a regular representation as the automorphism group of a k‐tournament. Our constructions are elementary, suggesting that the problem is significantly simpler for k‐tournaments than for binary tournaments. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 238–248, 2002