Equidistant frequency permutation arrays and related constant composition codes

In this paper we study the special class of equidistant constant composition codes of type CCC(n, d, μ ^m ) (where n = m μ), which correspond to equidistant frequency permutation arrays; we also consider related codes with composition “close to” μ ^m . We establish various properties of these objects and give constructions for optimal families of codes.     

Barely transitive permutation groups

Part of this work was done while the author was the recipient of RGC Grant 1423 awarded by the University of Alabama Research Grants Committee.     

On solvable minimally transitive permutation groups

We investigate properties of finite transitive permutation groups in which all proper subgroups of G act intransitively on . In particular, we are interested in reduction theorems for minimally transitive representations of solvable groups. Work partially supported by M.I.U.R. and London Mathematical Society.     

Reed-Muller codes and permutation decoding

We show that the first- and second-order Reed-Muller codes, R(1,m) and R(2,m), can be used for permutation decoding by finding, within the translation group, (m−1)- and (m+1)-PD-sets for R(1,m) for m≥5,6, respectively, and (m−3)-PD-sets for R(2,m) for m≥8. We extend the results of Seneviratne [P. Seneviratne, Partial permutation decoding for the first-order Reed-Muller codes, Discrete Math., 309 (2009), 1967-1970].     

Doubly transitive permutation groups with abelian stabilizers

Summary We prove that any doubly transitive permutation group with abelian stabilizers is the group of linear functions over a suitable field. The result is not new: for finite groups it is well known, for infinite groups it follows from a more general theorem of W. Kerby and H. Wefelscheid on sharply doubly transitive groups in which the stabilizers have finite commutator subgroups. We give a direct and elementary proof.     

The structure of transitive ordered permutation groups

We give some necessary and sufficient conditions for transitive l-permutation groups to be 2-transitive. We also discuss primitive components and give necessary and sufficient conditions for transitive l-permutation groups to be normal-valued.     

Endomorphism algebras of transitive permutation modules for p-groups

Let G be a finite p-group with subgroup H and k a field of characteristic p. We study the endomorphism algebra E = End_kG(k_H ↑^G), showing that it is a split extension of a nilpotent ideal by the group algebra kN_G(H)/H. We identify the space of endomorphisms that factor through a projective kG-module and hence the endomorphism ring of k_H ↑^G in the stable module category, and determine the Loewy structure of E when G has nilpotency class 2 and [G, H] is cyclic. Received: 3 November 2008     

A combinatorial approach to doubly transitive permutation groups

Let G be a doubly but not triply transitive group on a set X. We give an algorithm to construct the orbits of G acting on X×X×X by combining those of its stabilizer H of a point of X If the group H is given first, we compute the orbits of its transitive extension G, if it exists. We apply our algorithm to G=PSL(m,q) and Sp(2m,2), m3, successfully. We go forward to compute the transitive extension of G itself. In our construction we use a superscheme defined by the orbits of H on X×X×X and do not use group elements.     

On the order of doubly transitive permutation groups

Our aim is to contribute to an old problem of group theory. We prove that the order of a doubly transitive permutation group of degreen other thanA _n orS _n is less than exp exp . The best bound previously known was 4 ⁿ (published in 1980). The proof is based on results of A. Bochert (1892) and H. Wielandt (1934) and uses combinatorial techniques.Dedicated to Alfred Bochert and Helmut Wielandt     

On the point-stabiliser in a transitive permutation group

If V is a (possibly infinite) set, G a permutation group on ${V, v\in V}$ , and Ω is an orbit of the stabiliser G _v , let ${G_v^{\Omega}}$ denote the permutation group induced by the action of G _v on Ω, and let N be the normaliser of G in Sym(V). In this article, we discuss a relationship between the structures of G _v and ${G_v^\Omega}$ . If G is primitive and G _v is finite, then by a theorem of Betten et?al. (J Group Theory 6:415–420, 2003) we can conclude that every composition factor of the group G _v is also a composition factor of the group ${G_v^{\Omega(v)}}$ . In this paper we generalize this result to possibly imprimitive permutation groups G with infinite vertex-stabilisers, subject to certain restrictions that can be expressed in terms of the natural permutation topology on Sym(V). In particular, we show the following: If ${\Omega=u^{G_v}}$ is a suborbit of a transitive closed subgroup G of Sym(V) with a normalizing overgroup N?≤ N _Sym(V)(G) such that the N-orbital ${\{(v^g,u^g) \mid u\in \Omega, g\in N\}}$ is locally finite and strongly connected (when viewed as a digraph on V), then every closed simple section of G _v is also a section of ${G_v^\Omega}$ . To demonstrate that the topological assumptions on G and the simple sections of G _v cannot be omitted in this statement, we give an example of a group G acting arc-transitively on an infinite cubic tree, such that the vertex-stabiliser G _v is isomorphic to the modular group ${{\rm PSL}(2,\mathbb{Z}) \cong C_2*C_3}$ , which is known to have infinitely many finite simple groups among its sections.     

On extremely transitive extended perfect codes

We prove that, for every n = 2 ^k with k ≥ 4, there exist nonequivalent extremely transitive extended perfect codes. A transitive extended perfect code we call extremely transitive if the perfect code obtained from this code by puncturing any coordinate position is not transitive. The classification is given for all extended perfect codes of length 16.     

On the point-stabiliser in a transitive permutation group

If V is a (possibly infinite) set, G a permutation group on V, v ? V{V, v\in V}, and Ω is an orbit of the stabiliser G _v , let G_v^W{G_v^{\Omega}} denote the permutation group induced by the action of G _v on Ω, and let N be the normaliser of G in Sym(V). In this article, we discuss a relationship between the structures of G _v and G_v^W{G_v^\Omega}. If G is primitive and G _v is finite, then by a theorem of Betten et al. (J Group Theory 6:415–420, 2003) we can conclude that every composition factor of the group G _v is also a composition factor of the group G_v^W(v){G_v^{\Omega(v)}}. In this paper we generalize this result to possibly imprimitive permutation groups G with infinite vertex-stabilisers, subject to certain restrictions that can be expressed in terms of the natural permutation topology on Sym(V). In particular, we show the following: If W = u^G_v{\Omega=u^{G_v}} is a suborbit of a transitive closed subgroup G of Sym(V) with a normalizing overgroup N ≤ N _Sym(V)(G) such that the N-orbital {(v^g,u^g) | u ? W, g ? N}{\{(v^g,u^g) \mid u\in \Omega, g\in N\}} is locally finite and strongly connected (when viewed as a digraph on V), then every closed simple section of G _v is also a section of G_v^W{G_v^\Omega}. To demonstrate that the topological assumptions on G and the simple sections of G _v cannot be omitted in this statement, we give an example of a group G acting arc-transitively on an infinite cubic tree, such that the vertex-stabiliser G _v is isomorphic to the modular group PSL(2,\mathbbZ) @ C₂*C₃{{\rm PSL}(2,\mathbb{Z}) \cong C_2*C_3}, which is known to have infinitely many finite simple groups among its sections.