A 2-arc Transitive Hexavalent Nonnormal Cayley Graph on A 119

: A Cayley graph Γ = Cay ( G , S ) is said to be normal if the base group G is normal in Aut Γ . The concept of the normality of Cayley graphs was ﬁrst proposed by M.Y. Xu in 1998 and it plays a vital role in determining the full automorphism groups of Cayley graphs. In this paper, we construct an example of a 2-arc transitive hexavalent nonnormal Cayley graph on the alternating group A 119 . Furthermore, we determine the full automorphism group of this graph and show that it is isomorphic to A 120 .


Introduction
Throughout this paper, all graphs are assumed to be finite and undirected. For a graph Γ, we use VΓ, EΓ, ArcΓ and AutΓ to denote the vertex set, edge set, arc set and full automorphism group of the graph Γ, respectively. A graph Γ is said to be arc-transitive if the full automorphism group AutΓ acts transitively on ArcΓ. We use valΓ to denote the valency of the Γ, and we say Γ is a cubic, tetravalent, pentavalent or hexavalent graph, meaning valΓ = 3, 4, 5 or 6.
Let G be a finite group with identity element 1 and S (say Cayley subset) a subset of G such that 1 / ∈ S and S = S −1 := {x −1 | x ∈ S}. Define the Cayley graph Cay(G, S), that is, the Cayley graph of G with respect to the Cayley subset S as the graph with vertex set G such that g, h ∈ G are adjacent if and only if hg −1 ∈ S. It is easy to see that the valency of Cay(G, S) is |S|. As we all know, Cay(G, S) is connected if and only if S = G. On the other hand, letting R(G) be the right regular representation of G and letting AutCay(G, S) be the full automorphism group of Cay(G, S), there are clearly R(G) ≤ AutCay(G, S), and R(G) acts transitively on the vertices of Cay(G, S). Then, the graph Cay(G, S) is vertex-transitive, and G (or R(G)) can be viewed as a regular subgroup of AutCay(G, S). Conversely, a connected graph Γ is isomorphic to a Cayley graph of a group G if and only if the full automorphism group AutΓ contains a subgroup which acts regularly on VΓ and the subgroup is isomorphic to G (see [1]). A Cayley graph Γ = Cay(G, S) is said to be a normal Cayley graph if the base group G is normal in AutΓ; otherwise, Γ is said to be a nonnormal Cayley graph (see [2]).
The study about Cayley graphs on finite non-abelian simple groups has always attracted much attention because of Cayley graphs with high levels of symmetry; for example, vertex-transitivity, edge-transitivity and arc-transitivity are widely used in the design of interconnection networks. For more detailed applications, we recommend that readers refer to [3,4]. Let G be a finite non-abelian simple group, and let Γ = Cay(G, S) be a connected arc-transitive Cayley graph on G. The main motivation for classifying 2-arc-transitive nonnormal Cayley graphs comes from the fact that Fang, Ma and Wang [5] proved all but finitely that many locally primitive Cayley graphs of valency d ≤ 20 or a prime number of the finite non-abelian simple groups are normal. In [5] (Problem 1.2), they proposed the following problem: classify nonnormal locally primitive Cayley graphs (note that 2-arc-transitive graphs must be locally primitive) of finite simple groups with valency d ≤ 20 or a prime number. To solve this problem, we should study each valency d ≤ 20 or a prime number. In the case where Γ is a cubic graph (3-valent), Li [6] proved that Γ must be normal, except for seven exceptions. On the basis of Li's result, Xu et al. [7,8] proved that Γ must be normal, except for two exceptions on A 47 . In the case where Γ is a tetravalent graph (4-valent), Fang et al. in [9] proved that most of such Γ are normal, except for Cayley graphs on a list of G. Further, Fang et al. in [10] proved that Γ are normal when Γ is 2-transitive, except for two graphs on M 11 . In the case where Γ is a pentavalent graph (5-valent), Zhou and Feng [11] proved that all 1-transitive Cayley Γ of simple groups are normal. Ling and Lou in [12] gave an example of a 2-transitive pentavalent nonnormal Cayley graph on A 39 . Therefore, the next natural problem is to study the case of the 6-valent. However, there are no known nonnormal examples of hexavalent 2-arc-transitive Cayley graphs on finite simple groups.
The aim of this paper is to construct a nonnormal example of a connected 2-arc transitive hexavalent Cayley graph on a finite non-abelian simple group. Our main result is the following theorem. Theorem 1. There exists a nonnormal example of a connected 2-arc-transitive hexavalent Cayley graph on the alternating group A 119 , and the full automorphism group of this graph is isomorphic to the alternating group A 120 .

Preliminaries
In this section, we give some necessary preliminary results which are used in later discussions.
Let G be a finite group and let H be a subgroup of G. Then we have the following result (see [13] (Ch. I, 1.4)). We next introduce the definition of a Sabidussi coset graph. Let G be a group, g ∈ G\H such that g 2 ∈ H, and let H be a core-free subgroup of G. Define the Sabidussi coset graph Cos(G, H, g) of G with respect to the core-free subgroup H as the graph with vertex set [G : H] (the set of cosets of H in G) such that Hx and Hy are adjacent if and only if yx −1 ∈ HgH. The following lemma follows from [14], and it can be easily proved by the definition of the coset graphs (see [15] (Theorem 3) for example).

Lemma 2.
Let G be a group, g ∈ G\H such that g 2 ∈ H, and let H be a core-free subgroup of G. Let Γ = Cos(G, H, g) be a Sabidussi coset graph of G with respect to H. Then, Γ is G-arc-transitive and the following holds: (1) The valency of the graph Γ is equal to |H : Conversely, if Σ is an X-arc-transitive graph, then Σ is isomorphic to a Sabidussi coset graph Then we can verify that ϕ is a graph isomorphic from Σ to Cos(X, X v , g). Since Σ is undirected, we have g 2 ∈ X v . Hence, (H ∩ Hg) g = H ∩ Hg. Thus, we can choose a 2-element g satisfying g ∈ N X (X vw ).
Let t 1 ≥ 0 and t 2 ≥ 0 be two integers. We denote by the {2, 3}-group the finite group of the order 2 t 1 3 t 2 . Following the definition of relevant objects in [16] (Theorem 3.1), we have the following lemma, which is about the stabilizers of arc-transitive hexavalent graphs. For the structure of the received stabilizers, see the proof in [16] (Page 926). Lemma 3. Let s be a positive integer, and let Γ be a connected hexavalent (G, s)-transitive graph for some G ≤ AutΓ. Let v ∈ VΓ. Then s ≤ 4 and one of the following statements holds:

A 2-arc Transitive Hexavalent Nonnormal Cayley Graph on A 119
In this section, we construct a connected 2-arc transitive hexavalent nonnormal Cayley graph on A 119 and determine its full automorphism group. In fact, if Γ := Cay(G, S) is a Cayley graph of a non-abelian simple group G, then G is core free in X, where G ≤ X ≤ AutΓ. Let v ∈ VΓ and H = X v . Suppose that |H| = n. Then by Lemma 3, n may be 60, 120, etc. Consider the action of X on the set of [X : G] by right multiplication; then, X S n . So, we may construct the nonnormal Cayley graph in S n , where n = 60, 120, etc. The following example is really the case where we construct n = 120.  5 13 12 17)(6 14 11 18)(7 15 10 19)(8 16 9 20)(21 61 101 81 (4 37)(5 106)(6 105)(7 50)(8 49)(9 35)(10 45)(11 36 Proof. Let ∆ := {1, 2, . . . , 120}. Then, X has a natural action on ∆. By Magma [17], H, x = X, and so the graph Σ is connected by Lemma 2 (2). Furthermore, by Magma [17], we have that H is regular on ∆. However, G is the stabilizer of point 1 in X. Hence, X has a factorization X = GH = HG with G ∩ H = 1. Therefore, G is regular on [X : H]. By Lemma 2 (3), Σ is isomorphic to a Cayley graph of G = A 119 . Additionally, by the computation of Magma [17] (for the Magma code, see Appendix A), we have |H| |H∩H x | = 6. Hence, Lemma 2 (1) implies that Σ is a hexavalent graph. Since H ∼ = PGL(2, 5), Lemma 3 implies that Σ is 2-arc transitive. Since X is a non-abelian simple group, G is not normal in X ≤ AutΣ. It follows that Σ is nonnormal. Let x 1 , x 2 , x 3 , x 4 , x 5 and S be defined as in this lemma. By the computation of Magma [17] (for the Magma code, see Appendix B), we have G ∩ (HxH) = S. Thus, by Lemma 2 (3), we have that Σ is isomorphic to Cay(G, S) . This completes the proof of the lemma.
In the next lemma, we show that the full automorphism group AutΣ is isomorphic to alternating group A 120 . Lemma 5. The full automorphism group AutΣ of the 2-arc-transitive hexavalent graph Σ = Cos(X, H, x) in Construction 1 is isomorphic to alternating group A 120 .
Proof. Let A = AutΣ. Assume first that the full automorphism group A is quasiprimitive on VΣ. Let N be a minimal normal subgroup of A. Then, N is transitive on VΣ. It implies that N is insoluble. Thus, N is isomorphic to T 1 × T 2 × · · · × T d = T d , where T i ∼ = T for each 1 ≤ i ≤ d, T is a non-abelian simple group, and d ≥ 1. Let p be the largest prime factor of the order of A 119 . Then, p > 5 and p 2 |A 119 |. Since N is transitive on VΣ and |VΣ| = |A 119 |, we have that p divides |N|. Assume that d ≥ 2. Then, p d divides |N|. However, by Lemma 3, the order of the stabilizer A v divides 2 7 · 3 3 · 5 3 , and so |A| divides 2 7 · 3 3 · 5 3 · |A 119 | which is divisible by p d , a contradiction. Hence, we have d = 1 and N = T A. Let C = C A (T) be the centralizer of T in A. Then, C N A (T) = A and CT = C×T. If C = 1, since A is quasiprimitive on VΣ, this implies that C is transitive on VΣ. It implies that p divides |C|. Therefore, p 2 divides |CT|, which divides |A|, and so we have that p 2 divides |A|, a contradiction. Hence, C = 1, and A ≤ Aut(T) is almost simple.
Proof of Theorem 1. Now we are ready to prove our main Theorem 1. Let Σ = Cos(X, H, x) be the graph as in Construction 1. Then, Lemma 4 shows that Σ is a connected 2-arctransitive graph and isomorphic to a nonnormal hexavalent Cayley graph Cay(G, S), with G ∼ = A 119 . This proves the statement of the former part of Theorem 1. The next Lemma 5 shows that the full automorphism group AutΣ of the graph Σ is isomorphic to alternating group A 120 . This proves the statement of the latter part of Theorem 1, and so completes the proof of Theorem 1.