AbstractLetGbe a group acting symmetrically on a graph Σ, letG1be a subgroup ofGminimal among those that act symmetrically on Σ, and letG2be a subgroup ofG1maximal among those normal subgroups ofG1which contain no member except 1 which fixes a vertex of Σ. The most precise result of this paper is that if Σ has prime valencyp, then either Σ is a bipartite graph orG2acts regularly on Σ orG1|G2is a simple group which acts symmetrically on a graph of valencypwhich can be constructed from Σ and does not have more vertices than Σ. The results on vertex‐transitive groups necessary to establish results like this are also