Let D be a digraph, V (D) and A(D) will denote the sets of vertices and arcs of D, respectively. A digraph D is transitive if for every three distinct vertices u, v, w ∈ V (D), (u, v), (v, w) ∈ A(D) implies that (u,w) ∈ A(D). This concept can be generalized as follows: A digraph is k-transitive if for every u, v ∈ V (D), the existence of a uv-directed path of length k in D implies that (u, v) ∈...

A classi cation of all arc transitive graphs of order a product of two primes is given Furthermore it is shown that cycles and complete graphs are the only arc transitive Cayley graph of abelian group of odd order Introductory remarks Throughout this paper graphs are nite simple and undirected By p and q we shall always denote prime numbers A k arc in a graph X is a sequence of k vertices v v v...

The action of a subgroup G of automorphisms of a graph X is said to be 2 -transitive if it is vertexand edgebut not arc-transitive. In this case the graph X is said to be (G, 2)-transitive. In particular, X is 1 2 -transitive if it is (Aut X, 1 2)-transitive. The 2 -transitive action of G on X induces an orientation of the edges of X which is preserved by G. Let X have valency 4. An even length...

The groups in the title are classified, provided they are not too highly transitive. Let G be a primitive permutation group on a finite set S of n points. In 1871, Jordan initiated the study of G under the additional assumption that there is a transitive subgroup H of degree m, where 1 < m < n; that is, H fixes n m points and is transitive on the remaining points. THEOREM 1. I f m is a prime po...

A regular covering projection ℘: X̃ → X of connected graphs is G-admissible if G lifts along ℘. Denote by G̃ the lifted group, and let CT(℘) be the group of covering transformations. The projection is called G-split whenever the extension CT(℘) → G̃ → G splits. In this paper, split 2-covers are considered, with a particular emphasis given to cubic symmetric graphs. Supposing that G is transitive o...

In Zermelo-Fraenkel set theory with the axiom of choice every set has the same cardinal number as some ordinal. Von Rimscha has weakened this condition to “Every set has the same cardinal number as some transitive set.” In set theory without the axiom of choice, we study the deductive strength of this and similar statements introduced by von Rimscha. We shall use the standard notation and termi...

This paper forms part of a study of 2-arc transitivity for finite imprimitive symmetric graphs, namely for graphs admitting an automorphism groupG that is transitive on ordered pairs of adjacent vertices, and leaves invariant a nontrivial vertex partition B. Such a group G is also transitive on the ordered pairs of adjacent vertices of the quotient graph B corresponding toB. If in additionG is ...

Abstract We study some close relationships between the classes of transitive, fully transitive and Krylov torsion-free Abelian groups. In addition, as an application achieved assertions, we resolve old-standing problems, posed by Krylov, Mikhalev Tuganbaev in their monograph [P. A. V. Tuganbaev, Endomorphism Rings Groups, Kluwer Academic, Dordrecht, 2003]. Specifically, answer Problem 44 from t...

a subgroup x of a group g is said to be an h -subgroup if n_g(x) ∩ x^g ≤ x for each element g belonging to g. in [m. bianchi e. a., on finite soluble groups in which normality is a transitive relation, j. group theory, 3 (2000), 147–156] the authors showed that finite groups in which every subgroup has the h -property are exactly soluble groups in which normality is a transitive relation. here ...