CLASSIFICATION OF TETRAVALENT 2-TRANSITIVE NONNORMAL CAYLEY GRAPHS OF FINITE SIMPLE GROUPS

A graph Gamma is called (G, s)-arc-transitive if G <= Aut(Gamma) is transitive on the set of vertices of Gamma and the set of s-arcs of Gamma, where for an integer s >= 1 an s-arc of Gamma is a sequence of s + 1 vertices (v(0), v(1), ..., v(s)) of Gamma such that v(i-)(i) and v(i) are adjacent for 1 <= i <= s and # v(i-1) not equal v(i+1) for 1 <= i <= s - 1. A graph Gamma is called 2-transitive if it is (Aut(Gamma), 2)-arc-transitive but not (Aut(Gamma), 3)-arc-transitive. A Cayley graph Gamma of a group G is called normal if G is normal in Aut(Gamma ) and nonnormal otherwise. Fang et al. ['On edge transitive Cayley graphs of valency four', European J. Combin. 25 (2004), 1103-1116] proved that if Gamma is a tetravalent 2-transitive Cayley graph of a finite simple group G, then either Gamma is normal or G is one of the groups PSL2 (11), M-11, M-23 and A(11). However, it was unknown whether Gamma is normal when G is one of these four groups. We answer this question by proving that among these four groups only M-11 produces connected tetravalent 2-transitive nonnormal Cayley graphs. We prove further that there are exactly two such graphs which are nonisomorphic and both are determined in the paper. As a consequence, the automorphism group of any connected tetravalent 2-transitive Cayley graph of any finite simple group is determined.