On primitive permutation groups with small suborbits and their orbital graphs. (Q1880695)

In the present paper finite primitive permutation groups with a suborbit of length \(4\) or \(3\) are studied as well as vertex-primitive arc-transitive graphs of valency \(3\) or \(4\) and half-arc-transitive vertex-primitive graphs of valency \(2k\) for infinitely many integers \(k\geq 14\). Here a vertex-transitive graph is called half-arc-transitive if its automorphism group acts transitively on the edges but not transitively on the set of ordered pairs of adjacent vertices, called arcs. In particular, the authors establish a precise list of finite permutation groups with a suborbit of length \(4\) and show that the order of a point stabilizer in this case never is \(2^43^6\), contrary to a claim by \textit{W. L. Quirin} [Math. Z. 122, 267-274 (1971; Zbl 0213.03302)]. They also show that there are no half-arc-transitive graphs of valency less than \(10\).