Transitive Permutation Research Articles

Covers of complete graphs and related association schemes

We investigate the association schemes Inv(G) that are formed by the collection of orbitals of a permutation group G, for which the (underlying) graph Γ of a basis relation is a distance-regular antipodal cover of the complete graph. The group G can be regarded as an edge-transitive group of automorphisms of Γ and induces a 2-homogeneous permutation group on the set of its antipodal classes, which is either almost simple and 2-transitive, or affine. Using the classification of the finite 2-transitive permutation groups, we determine every possibility for Γ except for some special cases. This allows us to obtain a classification of such schemes Inv(G) provided that G is quasi-simple. We also give a general characterization of edge-transitive distance-regular antipodal covers of complete graphs in the almost simple case in terms of graphs of basis relations of schemes Inv(G) with quasi-simple group G. Then, we find several constructions of Deza graphs that are associated with such a scheme Inv(G). Finally, we establish isomorphisms between some graphs of basis relations of Inv(G) with G≃SU3(q) and abelian covers related to generalised quadrangles.

Circulant association schemes on triples

Association Schemes and coherent configurations (and the related Bose-Mesner algebra and coherent algebras) are well known in combinatorics with many applications. In the 1990s, Mesner and Bhattacharya introduced a three-dimensional generalisation of association schemes which they called an {\em association scheme on triples} (AST) and constructed examples of several families of ASTs. Many of their examples used 2-transitive permutation groups: the non-trivial ternary relations of the ASTs were sets of ordered triples of pairwise distinct points of the underlying set left invariant by the group; and the given permutation group was a subgroup of automorphisms of the AST. In this paper, we consider ASTs that do not necessarily admit 2-transitive groups as automorphism groups but instead a transitive cyclic subgroup of the symmetric group acts as automorphisms. Such ASTs are called {\em circulant} ASTs and the corresponding ternary relations are called {\em circulant relations}. We give a complete characterisation of circulant ASTs in terms of AST-regular partitions of the underlying set. We also show that a special type of circulant, that we call a {\em thin circulant}, plays a key role in describing the structure of circulant ASTs. We outline several open questions. 

Minimal binary 2-neighbour-transitive codes

The main result here is a characterisation of binary 2-neighbour-transitive codes with minimum distance at least 5 via their minimal subcodes, which are found to be generated by certain designs. The motivation for studying this class of codes comes primarily from their relationship to the class of completely regular codes. The results contained here yield many more examples of 2-neighbour-transitive codes than previous classification results of families of 2-neighbour-transitive codes. In the process, new lower bounds on the minimum distance of particular sub-families are produced. Several results on the structure of 2-neighbour-transitive codes with arbitrary alphabet size are also proved. The proofs of the main results apply the classification of minimal and pre-minimal submodules of the permutation modules over F2 for finite 2-transitive permutation groups.

The classification of \frac{3}2-transitive permutation groups and \frac{1}2-transitive linear groups

A linear group G ≤ G L ( V ) G\le GL(V) , where V V is a finite vector space, is called 1 2 \frac {1}{2} -transitive if all the G G -orbits on the set of nonzero vectors have the same size. We complete the classification of all the 1 2 \frac {1}{2} -transitive linear groups. As a consequence we complete the determination of the finite 3 2 \frac {3}{2} -transitive permutation groups – the transitive groups for which a point-stabilizer has all its nontrivial orbits of the same size. We also determine the ( k + 1 2 ) (k+\frac {1}{2}) -transitive groups for integers k ≥ 2 k\ge 2 .

The covering radii of the 2-transitive unitary, Suzuki, and Ree groups

Abstract We study the covering radii of 2-transitive permutation groups of Lie rank one, giving bounds and links to finite geometry.

Flag-transitive 4-(v, k, 3) designs and PSL(2, q) groups

Among the properties of homogeneity of incidence structures, flag-transitivity obviously is a particularly important and natural one. Originally, Buekenhout et al. reached a classification of flag-transitive Steiner 2-designs. Recently, Huber completely classified all flag-transitive Steiner t-designs with t ≤ 6 using the classification of the finite 2-transitive permutation groups. Hence the determination of all flag-transitive t-designs with λ ≥ 2 has remained of particular interest and has been known as a long-standing and still open problem.This article is a contribution to the study of the automorphism groups of 4-(v, k, 3) designs. Let S=(P,B) be a non-trivial 4-(q+1,k,3) design. If PSL(2, q) acts flag-transitively on S, then S is a 4-(168,12,3) design and GB is conjugate to A4 or Z12.

Arc-transitive distance-regular covers of cliques with λ = µ

We study antipodal distance-regular graphs of diameter 3 such that their automorphism group acts transitively on the set of pairs (a, b), where {a, b} is an edge of the graph. Since the automorphism group of such graphs acts 2-transitively on the set of antipodal classes, the classification of 2-transitive permutation groups can be used. We classify arc-transitive distance-regular graphs of diameter 3 in which any two vertices at distance at most two have exactly µ common neighbors.

Symmetric 2-Structures, a Classification

We classify symmetric 2-structures \({(P, \mathfrak{G}_1, \mathfrak{G}_2, \mathfrak{K})}\), i.e. chain structures which correspond to sharply 2-transitive permutation sets (E, Σ) satisfying the condition: “ \({(*) \, \, \forall \sigma, \tau \in \Sigma : \sigma \circ \tau^{-1} \circ \sigma \in \Sigma}\) ”. To every chain \({K \in \mathfrak{K}}\) one can associate a reflection \({\widetilde{K}}\) in K. Then (*) is equivalent to “ \({(**) \, \, \forall K \in \mathfrak{K} : \widetilde{K}(\mathfrak{K}) = \mathfrak{K}}\) ” and one can define an orthogonality “ \({\perp}\) ” for chains \({K, L \in \mathfrak{K}}\) by “ \({K \perp L \Leftrightarrow K \neq L \wedge \widetilde{K}(L) = L}\) ”. The classification is based on the cardinality of the set of chains which are orthogonal to a chain K and passing through a point p of K. For one of these classes (called point symmetric 2-structures) we proof that in each point there is a reflection and that the set of point reflections forms a regular involutory permutation set.

On cyclic semi-regular subgroups of certain 2-transitive permutation groups

We determine the cyclic semi-regular subgroups of the 2-transitive permutation groups PGL ( 2 , n ) , PSL ( 2 , n ) , PGU ( 3 , n ) , PSU ( 3 , n ) , Sz ( n ) and Ree ( n ) with n a suitable power of a prime number p .

Irreducible characters which are zero on only one conjugacy class

Suppose that G is a …nite solvable group which has an irreducible characterwhich vanishes on exactly one conjugacy class. Then we show that G has a homomorphic image which is a nontrivial 2-transitive permutation group. The latter groups have been classi…ed by Huppert. We can also say more about the structure of G depending on whether � is primitive or not. Mathematics Subject Classi…cation 2000: 20C15 20D10 20B20

The classification of flag-transitive Steiner 3-designs

Abstract We solve the long-standing open problem of classifying all 3-(v, k, 1) designs with a flag-transitive group of automorphisms (cf. [11], p. 147, and [12], p. 273, but presumably dating back to 1965). Our result relies on the classification of the finite 2-transitive permutation groups.

Irreducible and transitive locally-nilpotent by abelian finitary groups

We show that a locally-nilpotent by abelian group that is either a transitive finitary permutation group on an infinite set or an irreducible finitary skew linear group of infinite dimension, is a p-group for some suitably chosen prime p.

On minimal p-degrees in 2-transitive permutation groups

Suppose G is a transitive permutation group on a finite set \(\mit\Omega \) of n points and let p be a prime divisor of \(|G|\). The smallest number of points moved by a non-identity p-element is called the minimal p-degree of G and is denoted mp (G). ¶ In the article the minimal p-degrees of various 2-transitive permutation groups are calculated. Using the classification of finite 2-transitive permutation groups these results yield the main theorem, that \(m_{p}(G) \geq {{p-1} \over {p+1}} \cdot |\mit\Omega |\) holds, if \({\rm Alt}(\mit\Omega ) \nleqq G \).¶Also all groups G (and prime divisors p of \(|G|\)) for which \(m_{p}(G)\le {{p-1}\over{p}} \cdot |\mit\Omega |\) are identified.

Antipodal Distance Transitive Covers of Complete Graphs

A distance-transitive antipodal cover of a complete graphKnpossesses an automorphism group that acts 2-transitively on the fibres. The classification of finite simple groups implies a classification of finite 2-transitive permutation groups, and this allows us to determine all possibilities for such a graph. Several new infinite families of distance-transitive graphs are constructed.

Tiles for Concurrent and Located Calculi

When concurrency is a primitive notion, models of process calculi usually include commuting diamonds and observations of causality links or of abstract locations. However it is still debatable if the existing approaches are natural, or rather if they are an ad hoc addition to the more basic interleaving semantics. In the paper a treatment of concurrent process calculi is proposed where the same operational and abstract concurrent semantics described in the literature now descend from general, uniform notions. More precisely we introduce a tile-based semantics for located CCS and we show it consistent with the ordinary concurrent (via permutation of transitions) and bisimilarity based location semantics. Tiles are rewrite rules with side effects, reminiscent of both Plotkin SOS and Meseguer rewriting logic rules. We argue that the tile model is particularly well suited for defining directly operational and abstract semantics of concurrent process calculi in a compositional style.

The 2-transitive ovoids

An ovoid 0 in a classical polar space is a set of singular points such that every maximal totally singular subspace contains just one point in Lo. Ovoids are intimately connected with other combinatorial objects, including translation planes, spreads, partial geometries, codes, generalized hexagons, and Kerdock sets (see [S, 9, 163 for example). Interestingly enough, many of the known ovoids are in fact 2-transitive-that is, they admit a 2-transitive automorphism group (the notion of the automorphism group is made precise in (2.3)). For example, the Suzuki groups Sz(q) and the Ree groups *G,(q) act 2-transitively on ovoids in 4-dimensional symplectic geometry and 7-dimensional orthogonal geometry, respectively. Furthermore, the unitary groups PSU,(q) (for suitable prime powers q) and the linear groups PSL,(q3) (with q even) act 2-transitively on ovoids in 7or 8-dimensional orthogonal geometry. The occurrence of such a large number of 2-transitive ovoids suggests that a classification of them is worthwhile, much in the same spirit as Kantor’s classification of the finite linear spaces whose automorphism group acts 2-transitively on points [lo]. The classification of the 2-transitive ovoids appears as our Main Theorem in Section 2. We discover no new ovoids, however we obtain some new results concerning the number of isomorphism classes of 2-transitive ovoids (see Section 2). Our proof relies on the classification of the finite 2-transitive permutation groups, which in turn relies on the recent classification of finite simple groups. WC also draw upon several facts from the modular representation theory of finite groups.

Over the non-existence of sharply 3-transitive permutation sets containing sharply 2-transitive permutation subsets

The following results are proved: Let E be a finite set, ¦E¦>4, and let G be a sharply 3-transitive permutation set on E. Then G contains no subset which is a sharply 2-transitive permutation set on E (Theorem 1). In the case when G is a sharply 3-transitive permutation group which is also planar, the finiteness condition on E can be dropped (Theorem 2).

A new class of 2-transitive involutory permutation sets

A new class of 2-transitive involutory permutation sets

On projective planes of type (5, m)

If a group acts on a finite projective plane to make it a plane of type (5,m) then the related 2-transitive permutation representation of it has a normal regular subgroup.

Sharply triply transitive projective permutation groups and nonbinary codes generated by them

Sharply triply transitive projective permutation groups and nonbinary codes generated by them