Innately transitive subgroups of wreath products in product action

A permutation group is innately transitive if it has a transitive minimal normal subgroup, which is referred to as a plinth. We study the class of finite, innately transitive permutation groups that can be embedded into wreath products in product action. This investigation is carried out by observing that such a wreath product preserves a natural Cartesian decomposition of the underlying set. Previously we classified the possible embeddings in the case where the plinth is simple. Here we extend that classification and identify several different types of Cartesian decompositions that can be preserved by an innately transitive group with a non-abelian plinth. These different types of decompositions lead to different types of embeddings of the acting group into wreath products in product action. We also obtain a full characterisation of embeddings of innately transitive groups with diagonal type into such wreath products.

     

Arylthallium trifluoroacetates carbon-13 and fluorine-19 nuclear magnetic resonance: The substituent effect of the bis(trifluoroacetato)- thallium(III) group

¹³C NMR data for a series of arylthallium trifluoroacetates (ArTlX₂, X = OCOCF₃) are reported and assigned. The range of carbon—thallium couplings to be expected, the dependence on the disposition of coupled nuclei, and chemical shift effects are discussed. The Tl(OCOCF₃)₂ group is shown to be a powerful electron withdrawing group, from both the ¹³C data and ¹⁹F substituent chemical shifts of the p-fluorophenyl derivative.     

Locally s-distance transitive graphs and pairwise transitive designs

The study of locally s-distance transitive graphs initiated by the authors in previous work, identified that graphs with a star quotient are of particular interest. This paper shows that the study of locally s-distance transitive graphs with a star quotient is equivalent to the study of a particular family of designs with strong symmetry properties that we call nicely affine and pairwise transitive. We show that a group acting regularly on the points of such a design must be abelian and give general construction for this case.     

On the Automorphism Groups of Cayley Graphs of Finite Simple Groups

Let G be a finite nonabelian simple group and let be a connectedundirected Cayley graph for G. The possible structures for thefull automorphism group Aut are specified. Then, for certainfinite simple groups G, a sufficient condition is given underwhich G is a normal subgroup of Aut. Finally, as an applicationof these results, several new half-transitive graphs are constructed.Some of these involve the sporadic simple groups G = J₁, J₄,Ly and BM, while others fall into two infinite families andinvolve the Ree simple groups and alternating groups. The twoinfinite families contain examples of half-transitive graphsof arbitrarily large valency.     

Finding Optimal Routings in Hamming Graphs

A routing R in a graph consists of a simple path p_uvfromu to v for each ordered pair of distinct vertices (u, v). We will call R optimal if all the paths p_uvare shortest paths and if edges of the graph occur equally often in the paths of R. In 1994, Solé gave a sufficient condition involving the automorphism group for a graph to have an optimal routing in this sense. Graphs which satisfy Solé’s condition are called orbital regular graphs. It is often difficult to determine whether or not a given graph is orbital regular. In this paper, we give a necessary and sufficient condition for a Hamming graph to be orbital regular with respect to a certain natural subgroup of automorphisms.     

On Primitive Representations of Finite Alternating and Symmetric Groups with a 2-Transitive Subconstituent

In this paper we analyse primitive permutation representations of finite alternating and symmetric groups which have a 2-transitive subconstituent. We show that either the representation belongs to an explicit list of known examples, or the point stabiliser is a known almost-simple 2-transitive group and acts primitively in the natural representation of the associated alternating or symmetric group.     

On orbital partitions and exceptionality of primitive permutation groups

Let and be transitive permutation groups on a set such that is a normal subgroup of . The overgroup induces a natural action on the set of non-trivial orbitals of on . In the study of Galois groups of exceptional covers of curves, one is led to characterizing the triples where fixes no elements of ; such triples are called exceptional. In the study of homogeneous factorizations of complete graphs, one is led to characterizing quadruples where is a partition of such that is transitive on ; such a quadruple is called a TOD (transitive orbital decomposition). It follows easily that the triple in a TOD is exceptional; conversely if an exceptional triple is such that is cyclic of prime-power order, then there exists a partition of such that is a TOD. This paper characterizes TODs such that is primitive and is cyclic of prime-power order. An application is given to the classification of self-complementary vertex-transitive graphs.

     

Abelian coverings of finite general linear groups and an application to their non-commuting graphs

In this paper we introduce and study a family A_n(q)\mathcal{A}_{n}(q) of abelian subgroups of GL_n(q){\rm GL}_{n}(q) covering every element of GL_n(q){\rm GL}_{n}(q). We show that A_n(q)\mathcal{A}_{n}(q) contains all the centralizers of cyclic matrices and equality holds if q>n. For q>2, we obtain an infinite product expression for a probabilistic generating function for |A_n(q)||\mathcal{A}_{n}(q)|. This leads to upper and lower bounds which show in particular that