A circulant is a Cayley digraph over a finite cyclic group. The classification of arc-transitive circulants is shown. The result follows from earlier descriptions of Schur rings over cyclic groups.

Given natural numbers n ≥ 3 and 1≤ a, r ≤ n−1, the rose window graph R n (a, r) is a quartic graph with vertex set {x i |i ∈ Z n }∪{y i |i ∈ Z n } and edge set {{x i ,

As a generalization of undirected strongly regular graphs, a digraph X without loops, of valency k and order v is said to be a (v, k, μ, λ,t)-directed strongly regular graph whenever for any vertex u of X there are t undirected edges having u as an endvertex and for every two different vertices u and w of X the number of paths of length 2 starting at u and ending at w is λ or μ depending only on whether uw is an arc of X or not. An m-Cayley digraph of a group H is a digraph admitting a semiregular group of automorphisms having m orbits, all of equal length, isomorphic to H. In this paper, the structure of directed strongly regular 2-Cayley graphs of cyclic groups is investigated. In particular, the arithmetic conditions on parameters v, k, μ, λ, and t are given.Also, several infinite families of directed strongly regular graphs which are also 2-Cayley digraphs of abelian groups are constructed.

A finite group G is called a Schur group, if any Schur ring over G is associated in a natural way with a subgroup of Sym(G) that contains all right translations. Recently, the authors have completely identified the cyclic Schur groups. In this paper it is shown that any abelian Schur group belongs to one of several explicitly given families only. In particular, any non-cyclic abelian Schur group of odd order is isomorphic to Z 3 × Z 3 k or Z 3 × Z 3 × Z p where k ≥ 1 and p is a prime. In addition, we prove that Z 2 × Z 2 × Z p is a Schur group for every prime p.

A finite group G is called a Schur group if any Schur ring over G is associated in a natural way with a subgroup of Sym(G) that contains all right translations. It was proved by R. Pöschel (1974) that, given a prime p ≥ 5, a p-group is Schur if and only if it is cyclic. We prove that a cyclic group of order n is Schur if and only if n belongs to one of the following five families of integers: p k , pq k , 2pq k , pqr, 2pqr where p, q, r are distinct primes, and k ≥ 0 is an integer. §1. IntroductionLet G be a finite group. A subring of the group ring QG is called a Schur ring or S-ring over G if it is closed with respect to the componentwise multiplication and inversion. The first construction of such a ring was proposed by I. Schur [8] in connection with his famous result on permutation groups containing a regular cyclic subgroup. Namely, let Γ be a permutation group on the set G that contains the regular group G right induced by right multiplications, G right ≤ Γ ≤ Sym(G). Denote by Γ 1 the stabilizer of the identity of G in Γ. Then the submodule of QG spanned by the Γ 1 -orbits (transitivity module) is an S-ring over G. Such an S-ring was called schurian in [7]. The general theory of S-rings was developed by H. Wielandt in [9] where in particular he constructed an S-ring that cannot be obtained by the Schur method.