On the normality of Cayley digraphs of valency 2 on non-abelian groups of odd square free order

In this paper, we prove that all Cayley digraphs of valency 2 on nonabelian groups of odd square-free order are normal. For a given subset S of a finite group G without the identity element 1, the Cayley digraph on G with respect to S is denoted by r =Cay(G, S) where V(r) = G, E(r) = {(g, 8g) I 9 E G,8 E S}. It is clear that Aut (r), the automorphism group of r, contains the right regular representation G R of G as a subgroup. Moreover r is connected if and only if G = (S), and r is undirected if and only if S-l = S. r is called normal if G R is a normal subgroup of Aut (r). The concept of normality for Cayley digraphs is known to be important in the study of arc-transitive digraphs and half-tranisitive graphs. A natural problem is, for a given finite group G, to determine all normal or nonnormal Cayley digraphs of G. However this is a very difficult problem. The groups for which complete information about the normality of Cayley digraphs is available are cyclic groups of prime order (see [1]) and groups of order 2p (see [3]). Wang, Wang and Xu [9] determined all disconnected normal Cayley digraphs. Therefore we always suppose, in this paper, that the Cayley digraph Cay(G, S) is connected, that is, S is a generating subset of G. Xu [11, Problem 6] asked the following question: when S is a minimal generating set of G, are the corresponding Cayley digraph and graph normal? For abelian groups, Feng and Gao [5] proved that if the Sylow 2-subgroups of G are cyclic then the answers to the question are positive, and otherwise negative in general. About non abelian groups, Feng and XU [6J proved that there are only two nonnormal connected Cayley digraphs of valency 2 on nonabelian groups of order p3 and p4. This also implies that there are few nonnormal connected Cayley digraphs. Feng [4] determined all nonnormal Cayley digraphs of valency 2 on nonabelian groups of order 2p2. Wang and Li [10] also proved that the Cayley graphs of nonabelian groups *Supported by the National Natural Science Foundation of China (Grant no. 19671077) and Doctoral Program Foundation of the National Education Department of China Australasian Journal of Combinatorics 21(2000). pp.61-65 of order 2pq and of degree 2 are normal. In this paper we discuss the normality of connected Cayley digraphs of valency 2 on nonabelian groups of odd square-free order. Our result is the following: Main Theorem Let G be a nonabelian group of odd square-free order and let lSI = 2. Then r =Cay(G, S) is norma1. To prove our result, we need the following lemmas: Lemma 1 ([11, Prop. 1.5]) Let A = Aut (r) be the automorphism group of the Cayley digraph r of a group G with respect to its generating subset S and let Al be the stabilizer subgroup of A fixing the identity element 1 of G. Then r is normal if and only if Al is contained in the automorphism group Aut (G) of G. Lemma 2 ([4]) Let S = {e, f} be a two-generating subset of G without the identity 1 and let Ai be the subgroup of A which fixes the elements 1, e and f of G. Then r is normal if and only if Ai = 1. In this paper, we mainly discuss a normal subgroup A of the automorphism group of the Cayley digraph r = Cay( G, S) of valency 2 to determine whether r is normal. It is clear that IA : GI is a power of 2. To prove our theorem, we can assume that Cay(G, S) is not normal, where G is the smallest counterexample of odd square-free order. Let N be a smallest normal subgroup of A. Then N = TI X T2 X ... X Tk where Ti is isomorphic to Zp or a simple group. Since G is of odd square-free order, k = 1. When N is simple, since G is a Hall odd-subgroup of A, N n G is also a Hall odd-subgroup of N. Hence, by Corollary 5.6 of [2], N ~ PSL(2,p) where p is a Mersenne prime. Moreover, by Theorem II.8.27 of [7], G is the semidirect product of Zp by Z(p-I)/2' Now, we deal with the case when N is transitive on the set V(r) of the digraph r. Let (u, v) be a directed arc of r (the direction is from u to v). Then u and v are the tail and head of (u, v) respectively. If r has a circuit such that for every vertex u on this circuit, u is the tail of two incident arcs of the circuit or the head of two incident arcs, then the circuit is called an alternating circuit of r. Furthermore, if u is the tail of two incident arcs, then there exists at most one alternating circuit containing these two incident arcs; in which case we denote the circuit by O(u). Similarly if u is the head of two incident arcs of an alternating circuit we denote the circuit by J(u). Claim 3 In r, an alternating circuit must be an alternating cycle. Proof. When an alternating circuit A' of r is not an alternating cycle, there exist vertices which appear at least two times in A'. Since r is vertex-transitive and of valency 2, each vertex of A' must appear two times in AI. Hence, vertices not in A' are not adjacent to the vertices of A'. However, r is connected. Thus, all vertices appear in A'. Hence, the subgroup Ai, fixing A' pointwise, must fix all vertices of r. In other words, Ai = 1. By Lemma 2, r is normal. This is impossible. Now, we consider the alternating cycle construction of r. Since A is transitive, the length of the alternating cycles is a constant 2m where m is the number of vertices of valency 2 in an alternating cycle. Since Ai fixes the alternating cycle 0(1) pointwise, it must fix the set J((eI J)i) for 0 ~ i < m (see Figure 1 for m