C4C8(S) tori which are Cayley graphs

A [Formula: see text] net is a trivalent decoration made by alternating square [Formula: see text] and octagons [Formula: see text]. It can cover either a cylinder or a tori. Cayley graph [Formula: see text] on a group [Formula: see text] with connection set [Formula: see text] has the elements of [Formula: see text] as its vertices and an edge joining [Formula: see text] and [Formula: see text] for all [Formula: see text] and [Formula: see text]. Motivated by Afshari’s work, we show that the [Formula: see text] tori are Cayley graphs by constructing a regular subgroup of the automorphism group of [Formula: see text].

On the solvability of regular subgroups in the holomorph of a finite solvable group

We exhibit infinitely many natural numbers [Formula: see text] for which there exists at least one insolvable group of order [Formula: see text], and yet the holomorph of every solvable group of order [Formula: see text] has no insolvable regular subgroup. We also solve Problem 19.90(d) in the Kourovka notebook.

ON THE EIGENVALUES OF N-CAYLEY GRAPHS: A SURVEY

A graph Γ is called an n-Cayley graph over a group G if Aut(Γ) contains a semi-regular subgroup isomorphic to G with n orbits. In this paper, we review some recent results and future directions around the problem of computing the eigenvalues on n-Cayley graphs.

Permutation groups containing a regular abelian subgroup: the tangled history of two mistakes of Burnside

A group K is said to be a B-group if every permutation group containing K as a regular subgroup is either imprimitive or 2-transitive. In the second edition of his influential textbook on finite groups, Burnside published a proof that cyclic groups of composite prime-power degree are B-groups. Ten years later, in 1921, he published a proof that every abelian group of composite degree is a B-group. Both proofs are character-theoretic and both have serious flaws. Indeed, the second result is false. In this paper we explain these flaws and prove that every cyclic group of composite order is a B-group, using only Burnside’s character-theoretic methods. We also survey the related literature, prove some new results on B-groups of prime-power order, state two related open problems and present some new computational data.

Uniquely shift-transitive graphs of valency 5

An automorphism ? of a finite simple graph ? is a shift, if for every vertex v ? V(?),?v is adjacent to v in ?. The graph ? is shift-transitive, if for every pair of vertices u,v ? V(?) there exists a sequence of shifts ?1, ?2,...,?k 2 Aut(?) such that ?1?2...?ku = v. If, in addition, for every pair of adjacent vertices u,v ? V(?) there exists exactly one shift ? ? Aut(?) sending u to v, then ? is uniquely shift-transitive. The purpose of this paper is to prove that, if ? is a uniquely shift-transitive graph of valency 5 and S? is the set of shifts of ? then ?S??, the subgroup generated by S? is an Abelian regular subgroup of Aut(?).

Coset closure of a circulant S-ring and schurity problem

Let [Formula: see text] be a finite group. There is a natural Galois correspondence between the permutation groups containing [Formula: see text] as a regular subgroup, and the Schur rings (S-rings) over [Formula: see text]. The problem we deal with in the paper, is to characterize those S-rings that are closed under this correspondence, when the group [Formula: see text] is cyclic (the schurity problem for circulant S-rings). It is proved that up to a natural reduction, the characteristic property of such an S-ring is to be a certain algebraic fusion of its coset closure introduced and studied in the paper. Based on this characterization we show that the schurity problem is equivalent to the consistency of a modular linear system associated with a circulant S-ring under consideration. As a byproduct we show that a circulant S-ring is Galois closed if and only if so is its dual.

On non-abelian Schur groups

A finite group G is called Schur, if every Schur ring over G is associated in a natural way with a regular subgroup of sym (G) that is isomorphic to G. We prove that any non-abelian Schur group G is metabelian and the number of distinct prime divisors of the order of G does not exceed 7.

Cubic symmetric graphs of order twice an odd prime-power

An automorphism group of a graph is said to be s-regular if it acts regularly on the set of s-arcs in the graph. A graph is s-regular if its full automorphism group is s-regular. For a connected cubic symmetric graph X of order 2pn for an odd prime p, we show that if p ≠ 5, 7 then every Sylow p-subgroup of the full automorphism group Aut(X) of X is normal, and if p ≠3 then every s-regular subgroup of Aut(X) having a normal Sylow p-subgroup contains an (s − 1)-regular subgroup for each 1 ≦ s ≦ 5. As an application, we show that every connected cubic symmetric graph of order 2pn is a Cayley graph if p > 5 and we classify the s-regular cubic graphs of order 2p2 for each 1≦ s≦ 5 and each prime p. as a continuation of the authors' classification of 1-regular cubic graphs of order 2p2. The same classification of those of order 2p is also done.