Regular orbits for projective orthogonal groups over finite fields of odd characteristic

In this paper we prove that the projective orthogonal groups over finite fields of odd characteristic acting on the set of points of the corresponding quadrics, have regular orbits apart from a finite number of explicitly listed exceptions occurring in dimension 2 and 3.Lavoro eseguito nell'ambito dei gruppi nazionali del C.N.R. (G.N.S.A.G.A.) e del finanziamento del M.P.I.     

Functors between shape categories

Permutation groups of prime power degree are investigated here through the study of the corresponding group algebra of the set of all functions from the underlying set on which the permutation group acts to a finite field of characteristic p. For the case when the permutation group is of degree p² acting on a set consisting of the direct product of two elementary abelian p-groups, the structure of a minimal permutation module is obtained under certain conditions. The proofs do not depend on the recent classification results of finite simple groups.     

On permutation characters of wreath products

It is known that the character rings of symmetric groups S_n and the character rings of hyperoctahedral groups S₂?S_n are generated by (transitive) permutation characters. These results of Young are generalized to wreath products G?H (G a finite group, H a permutation group acting on a finite set). It is shown that the character ring of G?H is generated by permutation characters if this holds for G, H and certain subgroups of H. This result can be sharpened for wreath products G?S_n;if the character ring of G has a basis of transitive permutation characters, then the same holds for the character ring of G?S_n.     

Primitive <Emphasis Type="Italic">k</Emphasis>-free permutation groups

For a finite permutation group G acting on a set Ω, we say that G is k-free if the set-wise stabilizer of every k-subset of Ω is trivial. The purpose of this article is to describe, for all k, the primitive k-free permutation groups. Received: 20 February 2006     

Permutation binomials and their groups

This paper is devoted to studying the properties of permutation binomials over finite fields and the possibility to use permutation binomials as encryption functions. We present an algorithm for enumeration of permutation binomials. Using this algorithm, all permutation binomials for finite fields up to order 15000 were generated. Using this data, we investigate the groups generated by the permutation binomials and discover that over some finite fields \mathbb F_q {{\mathbb F}_q} , every bijective function on [1..q − 1] can be represented as a composition of binomials. We study the problem of generating permutation binomials over large prime fields. We also prove that a generalization of RSA using permutation binomials is not secure. Bibliography: 9 titles.     

Conjugacy classes of derangements in finite transitive groups

Let G be a permutation group acting transitively on a finite set Ω. We classify all such (G, Ω) when G contains a single conjugacy class of derangements. This was done under the assumption that G acts primitively by Burness and Tong-Viet. It turns out that there are no imprimitive examples. We also discuss some results on the proportion of conjugacy classes which consist of derangements.     

The Permutation Modules for Finite (Affine) General Linear Groups

The permutation representations of finite general linear and affine groups on the set of vectors of their standard module are studied. The permutation modules over fields and local rings are considered. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 13, Algebra, 2004.     

Permutation polynomials of degree 6 or 7 over finite fields of characteristic 2

In Dickson (1896–1897) [2], the author listed all permutation polynomials up to degree 5 over an arbitrary finite field, and all permutation polynomials of degree 6 over finite fields of odd characteristic. The classification of degree 6 permutation polynomials over finite fields of characteristic 2 was left incomplete. In this paper we complete the classification of permutation polynomials of degree 6 over finite fields of characteristic 2. In addition, all permutation polynomials of degree 7 over finite fields of characteristic 2 are classified.     

Extension of Multiply Transitive Permutation Sets

Let G be a k-transitive permutation set on E and let E* = E∪{∞},∞ ? E; if G* is a (k: + 1)-transitive permutation set on E*, G* is said to be an extension of G whenever G _∝ ^* =G. In this work we deal with the problem of extending (sharply) k- transitive permutation sets into (sharply) (k + 1)-transitive permutation sets. In particular we give sufficient conditions for the extension of such sets; these conditions can be reduced to a unique one (which is a necessary condition too) whenever the considered set is a group. Furthermore we establish necessary and sufficient conditions for a sharply k- transitive permutation set (k ≥ 3) to be a group. Math. Subj. Class.: 20B20 Multiply finite transitive permutation groups 20B22 Multiply infinite transitive permutation groups     

Graphical Frobenius representations

A Frobenius group is a transitive permutation group that is not regular and such that only the identity fixes more than one point. A graphical Frobenius representation (GFR) of a Frobenius group G is a graph whose automorphism group, as a group of permutations of the vertex set, is isomorphic to G. The problem of classifying which Frobenius groups admit a GFR is a natural extension of the classification of groups that have a graphical regular representation (GRR), which occupied many authors from 1958 through 1982. In this paper, we review for graph theorists some standard and deep results about finite Frobenius groups, determine classes of finite Frobenius groups and individual groups that do and do not admit GFRs, and classify those Frobenius groups of order at most 300 having a GFR. Because a Frobenius group, as opposed to a regular permutation group, has a highly restricted structure, the GFR problem emerges as algebraically more complex than the GRR problem. This paper concludes with some further questions and a strong conjecture.     

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.     

Symmetries of directed graphs and the Chinese remainder theorem

It is shown that a permutation group on a finite set is the automorphism group of some directed graph if and only if a generalized Chinese remainder theorem holds for the family of stabilizers. This result can be applied to examine some special permutation groups, including the general linear groups of finite vector spaces.     

On graphical regular representations of direct products of groups

A given group G may or may not have the property that there exists a graph X such that the automorphism group of X is regular, as a permutation group, and isomorphic to G. Mark E. Watkins has shown that the direct product of two finite groups has this property if each factor has this property and both factors are different from the cyclic group of order 2. Later, Wilfried Imrich generalized this result to infinite groups. In this paper, a new proof of this result for finite groups is given. The proof rests heavily on the result which states that if X is a graphical regular representation of the group G, then X is not self-complementary.     

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.      

Circular Tuscan-k Arrays from Permutation Binomials

In this paper, we first discuss some properties of permutation polynomials over finite fields. In particular, a class of permutation binomials are introduced and a series of set complete mappings is constructed. Based on that, we present a new construction for Tuscan-l arrays with various sizes.     

Permutation polynomials and orthomorphism polynomials of degree six

A classic paper of Dickson gives a complete list of permutation polynomials of degree less than 6 over arbitrary finite fields, and degree 6 over finite fields of odd characteristic. However, some published statements have hinted that Dicksonʼs classification might be incomplete in the degree 6 case. We uncover the reason for this confusion, and confirm the list of degree 6 permutation polynomials over all finite fields. Using this classification, we determine the complete list of degree 6 orthomorphism polynomials. Additionally, we note that a family of permutation polynomials from Dicksonʼs list provides counterexamples to a published conjecture of Mullen.     

Constructions for Permutation Codes in Powerline Communications

A permutation array (or code) of length n and distance d is a set Γ of permutations from some fixed set of n symbols such that the Hamming distance between each distinct x, y ∈ Γ is at least d. One motivation for coding with permutations is powerline communication. After summarizing known results, it is shown here that certain families of polynomials over finite fields give rise to permutation arrays. Additionally, several new computational constructions are given, often making use of automorphism groups. Finally, a recursive construction for permutation arrays is presented, using and motivating the more general notion of codes with constant weight composition.     

The Closures of Wreath Products in Product Action

Algebra and Logic - Let m be a positive integer and let Ω be a finite set. The m-closure of G ≤ Sym(Ω) is the largest permutation group G(m) on Ω having the same orbits as G in...     

Two-Arc Transitive Graphs and Twisted Wreath Products

The paper addresses a part of the problem of classifying all 2-arc transitive graphs: namely, that of finding all groups acting 2-arc transitively on finite connected graphs such that there exists a minimal normal subgroup that is nonabelian and regular on vertices. A construction is given for such groups, together with the associated graphs, in terms of the following ingredients: a nonabelian simple group T, a permutation group P acting 2-transitively on a set , and a map F : Tsuch that x = x ^–1 for all x F() and such that Tis generated by F(). Conversely we show that all such groups and graphs arise in this way. Necessary and sufficient conditions are found for the construction to yield groups that are permutation equivalent in their action on the vertices of the associated graphs (which are consequently isomorphic). The different types of groups arising are discussed and various examples given.