Some remarks on orbital digraphs for the finite primitive groups

Arxiv preprint math/0611758, 2006

If G is a group acting on a set V and a, b are elements of V, the digraph whose vertex set is V and whose arc set is the orbit (a, b)^G is called an orbital digraph of G. A locally finite digraph X has more than one end if there exists a finite set of vertices Y such that the induced digraph X \ Y contains at least two infinite connected components; if there exists such a set containing precisely one element, then X has connectivity one. In this paper we show that if G is a primitive permutation group whose suborbits are all finite, possessing an orbital digraph with more than one end, then G has a primitive connectivity-one orbital digraph, and this digraph is essentially unique. Such digraphs resemble trees in many respects, and have been fully characterised in another paper (Infinite primitive digraphs) by the author.

The authors present a natural generalization of orbital graphs. If Γ is a subgroup of S n ×S n , V an n-set, and (u,v)∈V×V, then the set Γ(u,v)={(α(u),β(v))∣(α,β)∈Γ} forms the arc set of a digraph G with vertex set V; G is called a two-fold orbital digraph. The authors focus on properties of such a G which distinguish this class from that of orbital graphs (the undirected orbital case). Special attention is given to the case where G is disconnected and to the relationship between G and its canonical double covering.

Journal of Combinatorial Theory, Series B, 1981

Suppose that G is a simply transitive primitive permutation group on a finite set such that for a in ~ the stabilizer G a is 2-transitive on one of its orbits F(a) in .Q of length v > 2. Peter J. Cameron showed that G,~ has another orbit related to F(a) of length v(v-1)/l~ 2v. Here we show that under certain extra, fairly natural, conditions (including the assumptions that G,~ is not faithful on F(a)), G,~ has a third orbit of length greater than v(v-I), and obtain strong restrictions on the length of this orbit when G~ is more than 2-transitive on F(a). In fact if G rt'~) ~_A v and G o is not faithful on F(a) we show that either the third orbit has length V(V-1) 2, or G is S2v_ ~ or A2v_ 1 acting as an automorphism group of the odd graph 0~, where the set of points is identified with O and the set of points adjacent to a is precisely F(a). Suppose that G is a simply transitive permutation group on a finite set ~O such that for a in .(2 the stabilizer G~ of a acts 2-transitively on one of its orbits F(a) in .O, where IF(a)l = v > 2. Under these assumptions Peter J. Cameron (generalizing an old result of Manning [13]), showed that F o F*(a) = {a; fl E ~2, F(a) ~ F(fl) ve 4, a :/: fl} is an orbit of G~ of length v(v-1)/l>~ 2v, and found stronger restrictions on l when G rt~) is more highly transitive (see [3, 4]). Also under these assumptions Knapp [12], and B/irker and Knapp [2] obtained information about the structure of G~. In this paper we assume that G,~ does not act faithfully on F(a) and, using a combination of the methods of Cameron, and Bfirker and Knapp, we investigate a third orbit of G,~. First we formalise our assumptions and notation. Hypothesis A. G is a simply transitive primitive permutation group on a finite set .(2. For a E Q, G~ has an orbit F(a) in .Q of length IF(a)[ = v > 2 on which it is 2-transitive and not faithful. Notation. (a) If G is a permutation group on .Q there is a natural action Of G on the cartesian product ~2 • .

We consider the problem of determining the maximum possible out-degree d(n) of a digraph on n vertices which admits a sharply edge-transitive group. We show that d(n) > en/log log n for every n, while d(n) = in infinitely often. Also, d(n) = n -1 if and only if n is a prime power, whereas for non-prime-power values of n, we show that nd(n) tends to infinitely with n. The question has interesting grouptheoretic aspects. This and related problems generalise the existence question for projective planes.

In this paper G denotes a finite p-group and Γ denotes a simple undirected graph. The orbit graph is a graph whose vertices are non-central orbit under group action of G on a set. Two vertices v1 and v2 are adjacent in the graph if vgv12= where vv12,,∈Ω and gG∈. In this paper, the orbit graph of some finite p-groups and group of order pq, where p and q are relatively prime, is found. The orbit graph is determined for the group in the case that a group acts regularly on itself, acts on itself by conjugation, and acts on a set. Besides, some graph properties are found.