VoL 44 No. I SCIENCE IN CHINA (Series A) January 2001 Locally primitive Cayley graphs of finite simple groups FANG Xingui (~,~-)1, C. E. Praeger 2 & WANG Jie (_.T.. ,~,,)l l. Department of Mathematics, Peking University, Beijing 100871, China 2. Department of Mathematics, The University of Western Australia Perth, WA 6907, Australia Correspondence should be addressed to Fang Xingui Received June 3, 2000 Abstract A graph /- is said to be G-locally primitive, where G is a subgroup of automorphisms of /-, if the stabiliser G~ of a vertex a acts primitively on the set/-(a) of vertices of/- adjacent to or. For a finite non-abelian simple group L and a Cayiey subset S of L, suppose that L <3 G...<Aut( L), and the Cayley graph /- = Cay ( L, S) is G-locally primitive. In this paper we prove that L is a simple group of Lie type, and either the valency of /- is an add prine divisor of I Out(L) I, or L = PQ{ (q) and/- has valency 4. In either cases, it is proved that the full automorphism group of/- is also almost simple with the same socle L. Keywords: finite simple group, Cayley graph, locally primitive, quasiprimitive, semiregular. For a group G, its socle soc (G) is the product of all minimal normal subgroups of G. A group G is said to be almost simple if its socle L: = soc(G) is a nonabelian simple group and hence L <3 G ~< Aut( L ). A transitive group G is said to be quasiprimitive if each nontrivial nor- mal subgroup is transitive. A permutation group G is semiregular if only the identity element has fixed point. If G is both transitive and semiregular we say that G is regular. The arcs of a graph P are the ordered pairs of adjacent vertices. If a group G acts as a transitive group of automor- phisms of a graph/-" then F' is said to be G-locally primitive if, for all vertices a, the vertex sta- biliser G~ acts primitively on the set /-'( a ) of vertices adjacent to a. In refs. [ 1,2] an investigation was begun into finite connected nonbipartite graphs P ad- mitting an almost simple subgroup G of automorphisms such that G is transitive on the arcs of/1, G is locally primitive on F, and G is quasiprimitive on vertices of/-'. The aim was to deduce, from these properties of the subgroup G, certain properties of the full automorphism group Aut (r') of F, for example, to determine part of the subgroup structure of Aut(/-') if Aut(/-') is not almost simple, or if Aut(/') is not quasiprimitive on vertices. Unfortunately the results in refs. [ 1,2 ] showed as a hypothesis that the simple socle L of G was not semiregular on vertices. The purpose of this paper is to examine the case where L is semiregular on vertices to determine the extent to which the results in refs. [ 1,2 ] depend on this assumption. It turns out (see Theorem 1 ) that in this situation, L is a simple group of Lie type, F is a Cayley graph for L with valency either an odd prime or 4, and the full automorphism group Aut (/-') also has socle L. Thus, with G, L, /-" as in the above paragraph, we assume that L is semiregular on ver- tices. In this case, since G is quasiprimitive on vertices, L is transitive, and hence regular on vertices. This means that /-' is a Cayley graph for L (see Lemma 16.3 of ref. [ 3 ] ). Cayley graphs are defined as follows. For a group L, and a subset S of L such that 1 ~ S and S = S - 1, the Cayley graph/-" = Cay ( L, S) of L with respect to S is defined as the graph having