The Infinitude of 7-Arc-Transitive Graphs

By theorems of Tutte, Weiss, and others, it is known that there are no finite symmetric graphs of degree greater than 2 with automorphism group transitive on 8-arcs, and that 7-arc-transitivity can occur only in the case of graphs of degree 3m + 1. In this article it is shown that there are infinitely many 7-arc-transitive finite quartic graphs; indeed for all but finitely many positive integersn, there is a finite connected 7-arc-transitive quartic graph with the alternating groupAnacting transitively on its 7-arcs, and another with the symmetric groupSnacting transitively on its 7-arcs. The proof uses a construction involving permutation representations of a generic infinite group to produce an infinite family of finite graphs with the required properties.