Distance Transitive Generalized Quadrangles of Prime Order

Without using the classification of finite simple groups, we classify the finite generalized quadrangles of prime order admitting a group acting distance transitively on the collinearity graph. Our method uses combinatorial geometry and permutation groups.     

Local Sharply Transitive Actions on Finite Generalized Quadrangles

We classify the finite generalized quadrangles containing a line L such that some group of collineations acts sharply transitively on the ordered pentagons which start with two points of L. This can be seen as a generalization of a result of Thas and the second author [22] classifying all finite generalized quadrangles admitting a collineation group that acts transitively on all ordered pentagons, although the restriction to sharp transitivity is essential in our arguments. However, the conclusion is exactly the same family of classical generalized quadrangles (the orthogonal quadrangles and their duals). Our main result thus provides a local group theoretic characterization of these classical quadrangles.     

Generalized Quadrangles with an Abelian Singer Group

In this note we characterize thick finite generalized quadrangles constructed from a generalized hyperoval as those admitting an abelian Singer group, i.e., an abelian group acting regularly on the points. S. De Winter: The first author is a Research Assistant of the Fund for Scientific Research—Flanders (Belgium). K. Thas: The second author is a Postdoctoral Fellow of the Fund for Scientific Research—Flanders (Belgium).     

<Emphasis Type="Italic">k</Emphasis>–Pyramidal One–Factorizations

We consider one–factorizations of complete graphs which possess an automorphism group fixing k ≥ 0 vertices and acting regularly (i.e., sharply transitively) on the others. Since the cases k = 0 and k = 1 are well known in literature, we study the case k≥ 2 in some detail. We prove that both k and the order of the group are even and the group necessarily contains k − 1 involutions. Constructions for some classes of groups are given. In particular we extend the result of [7]: let G be an abelian group of even order and with k − 1 involutions, a one–factorization of a complete graph admitting G as an automorphism group fixing k vertices and acting regularly on the others can be constructed.     

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     

A new result on difference sets with -1 as multiplier

In this paper we study finite abelian groups admitting a difference set with multiplier -1. In these groups we have that each integer, which is relatively prime to the group order, is a multiplier (see [1] and Section 1 of this paper).About the arithmetical structure, there is an interesting result of Jungnickel [3] on primes dividing the order n of the corresponding design. Here we prove (see Theorem 2.1) that each odd prime divisor of the order v of the group divides n. The proof of Theorem 2.1 rests on character computations and is motivated by [5].     

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     

Correction: Generalized Polygons with Highly Transitive Collineation Groups

In Geom Dedicata 58 (1995), 91–100, the author tried to classify generalized quadrangles with a collineation group acting transitively on ordered pentagons. Unfortunately, this paper contains several mistakes. The main result is affected such that there now is an additional compact connected quadrangle (and its dual) with a 5-gon transitive group. I am indebted to Martin Wolfrom who pointed out the error and offered a correction.     

Regular groups on generalized quadrangles and nonabelian difference sets with multiplier -1

This is a first approach to the study of regular generalized quadrangles (i.e. generalized quadrangles with an automorphism group sharply 1-transitive on points). In this paper we point out how the problem is connected to the theory of difference sets with multiplier-1. First, some of the results in [3] on difference sets with multiplier-1 are extended to the nonabelian case; then, these new results on difference sets are used to prove nonexistence theorems for regular GQs of even order s=t.Dedicated to Otto Wagner on the occasion of his 60th birthday     

On the Artin exponent of some rational groups

A finite group G is called rational if all its irreducible complex characters are rational valued. In this paper, we show that if G is a direct product of finitely many rational Frobenius groups then every rationally represented character of G is a generalized permutation character. Also we show that the same assertion holds when G is a solvable rational group with a Sylow 2-subgroup isomorphic to the dihedral group of order 8 and an abelian normal Sylow 3-subgroup.     

The linking pairings of orientable Seifert manifolds

We compute the p-primary components of the linking pairings of orientable 3-manifolds admitting a fixed-point free S¹-action. Any linking pairing on a finite abelian group of odd order is realized by such a manifold. We find necessary and sufficient conditions for a pairing on an abelian 2-group to be the 2-primary component of such a linking pairing, and give simple examples which are not realizable by any Seifert fibred 3-manifold.     

Central aspects of skew translation quadrangles, 1

Modulo a combination of duality, translation duality or Payne integration, every known finite generalized quadrangle except for the Hermitian quadrangles \(\mathcal {H}(4,q^2)\), is an elation generalized quadrangle for which the elation point is a center of symmetry—that is, is a “skew translation generalized quadrangle” (STGQ). In this series of papers, we classify and characterize STGQs. In the first installment of the series, (1) we obtain the rather surprising result that any skew translation quadrangle of finite odd order (s, s) is a symplectic quadrangle; (2) we determine all finite skew translation quadrangles with distinct elation groups (a problem posed by Payne in a less general setting); (3) we develop a structure theory for root elations of skew translation quadrangles which will also be used in further parts, and which essentially tells us that a very general class of skew translation quadrangles admits the theoretical maximal number of root elations for each member, and hence, all members are “central” (the main property needed to control STGQs, as which will be shown throughout); and (4) we show that finite “generic STGQs,” a class of STGQs which generalizes the class of the previous item (but does not contain it by definition), have the expected parameters. We conjecture that the classes of (3) and (4) contain all STGQs.     

Quaternionic Starters

Let m be an integer, m 2 and set n = 2^m. Let G be a non-cyclic group of order 2n admitting a cyclic subgroup of order n. We prove that G always admits a starter and so there exists a one–factorization of K_2n admitting G as an automorphism group acting sharply transitively on vertices. For an arbitrary even n > 2 we also show the existence of a starter in the dicyclic group of order 2n.Research performed within the activity of INdAM–GNSAGA with the financial support of the Italian Ministry MIUR, project Strutture Geometriche, Combinatoria e loro Applicazioni     

Isomorphisms of groups related to flocks

A truly fruitful way to construct finite generalized quadrangles is through the detection of Kantor families in the general 5-dimensional Heisenberg group over some finite field $\mathbb{F}_{q}$ . All these examples are so-called ??flock quadrangles??. Payne (Geom. Dedic. 32:93?C118, 1989) constructed from the Ganley flock quadrangles the new Roman quadrangles, which appeared not to arise from flocks, but still via a Kantor family construction (in some group of the same order as ). The fundamental question then arose as to whether (Payne in Geom. Dedic. 32:93?C118, 1989). Eventually the question was solved in Havas et?al. (Finite geometries, groups, and computation, pp.?95?C102, de Gruyter, Berlin, 2006; Adv. Geom. 26:389?C396, 2006). Payne??s Roman construction appears to be a special case of a far more general one: each flock quadrangle for which the dual is a translation generalized quadrangle gives rise to another generalized quadrangle which is in general not isomorphic, and which also arises from a Kantor family. Denote the class of such flock quadrangles by . In this paper, we resolve the question of Payne for the complete class . In fact we do more??we show that flock quadrangles are characterized by their groups. Several (sometimes surprising) by-products are described in both odd and even characteristic.     

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     

The permutation action of finite symplectic groups of odd characteristic on their standard modules

We study the space of functions on a finite-dimensional vector space over a field of odd order as a module for a symplectic group. We construct a basis of this module with the following special properties. Each submodule generated by a single basis element under the symplectic group action is spanned as a vector space by a subset of the basis and has a unique maximal submodule. From these properties, the dimension and composition factors of the submodule generated by any subset of the basis can be determined. These results apply to incidence geometry of the symplectic polar space, yielding the symplectic analogue of Hamada's additive formula for the p-ranks of the incidence matrices between points and flats. A special case leads to a closed formula for the p-rank of the incidence matrix between the points and lines of the symplectic generalized quadrangle over a field of odd order. Together with earlier results on the 2-ranks, this result completes the determination of the p-ranks for these quadrangles.     

On Schurity of Finite Abelian Groups

A finite group G is called a Schur group, if any Schur ring over G is associated in a natural way with a subgroup of Sym(G) that contains all right translations. Recently, the authors have completely identified the cyclic Schur groups. In this article, it is shown that any abelian Schur group belongs to one of several explicitly given families only. In particular, any noncyclic abelian Schur group of odd order is isomorphic to ?₃ × ?_{3 ^k} or ?₃ × ?₃ × ? _p where k ≥ 1 and p is a prime. In addition, we prove that ?₂ × ?₂ × ? _p is a Schur group for every prime p.     

Groups Admitting a Kantor Family and a Factorized NormalSubgroup

We study the structure of a finite groupG admitting a Kantor family (F, F ^*) of type (s, t) and a nontrivial normal subgroupX which isfactorized byF F ^*. The most interesting cases, giving necessary conditions on the structure ofG and the parameterss andt, are those where a further Kantor family is induced inX, or where a partial congruence partition is induced in the factor groupG/X. Most of the known finite generalized quadrangles can be constructed as coset geometries with respect to a Kantor family. We show that the parameters of a skew translation generalized quadrangle necessarily are powers of the same prime. Furthermore, the structure of nonabelian groups admitting a Kantor family consisting only of abelian members is considered.Dedicated to Hanfried Lenz on the occasion of his 80th birthday     

Fusions of Character Tables II: p-Groups

If the character table of a finite group H satisfies certain conditions, then the classes and characters of H can fuse to give the character table of a group G of the same order. We investigate the case where H is an abelian group. In a previous article, we gave examples of Camina pairs that fuse from abelian groups. In this article, we give more general examples of Camina triples that fuse from abelian groups. We use this result to give an example of a group which fuses from an abelian group, but which has a subgroup that does not. We also give an example of a powerful 2-group which does not fuse from an abelian group and of a regular 3-group which does not fuse from an abelian group.     

On infinite tournaments with regular automorphism groups

Atournament regular representation (TRR) of an abstract groupG is a tournamentT whose automorphism group is isomorphic toG and is a regular permutation group on the vertices ofT. L. Babai and W. Imrich have shown that every finite group of odd order exceptZ ₃ ×Z ₃ admits a TRR. In the present paper we give several sufficient conditions for an infinite groupG with no element of order 2 to admit a TRR. Among these are the following: (1)G is a cyclic extension byZ of a finitely generated group; (2)G is a cyclic extension byZ _2n+1 of any group admitting a TRR; (3)G is a finitely generated abelian group; (4)G is a countably generated abelian group whose torsion subgroup is finite.