Pentavalent arc-transitive Cayley graphs on Frobenius groups with soluble vertex stabilizer

Abstract A Cayley graph Γ is said to be arc-transitive if its full automorphism group AutΓ is transitive on the arc set of Γ. In this paper we give a characterization of pentavalent arc-transitive Cayley graphs on a class of Frobenius groups with soluble vertex stabilizer.


Introduction
Throughout the paper, graphs considered are simple, connected and undirected. For a graph Γ, we denote the vertex set, edge set, arc set, valency and full automorphism group of Γ by VΓ, EΓ, AΓ, val(Γ) and AutΓ, respectively. Γ is said to be G-vertex-transitive, G-edge-transitive or G-arc-transitive if G ≤ AutΓ is transitive on VΓ, EΓ or AΓ; in particular, if G = AutΓ, then Γ is simply called vertex-transitive, edge-transitive or arctransitive. An s-arc of Γ is a sequence of vertices (u , u , . . . , us) such that u i is adjacent to u i+ and u i− ≠ u i+ for all possible i. For a subgroup G AutΓ, Γ is said to be (G, s)-arc-transitive if G is transitive on the set of s-arcs in Γ. In particular, a -arc is called a vertex, and a -arc is called an arc for short. As we all know, a graph Γ is G-arc-transitive for some G ≤ AutΓ if and only if G is transitive on VΓ and the vertex stabilizer Gv of v ∈ VΓ in G is transitive on the neighborhood Γ(v) of v.
A graph Γ is called a Cayley graph if there exist a group G and a subset S ⊂ G \ { } with S = S − : = {g − | g ∈ S} such that the vertices of Γ may be identi ed with the elements of G in such a way that x is adjacent to y if and only if yx − ∈ S. The Cayley graph Γ is denoted by Cay(G, S). Throughout this paper, we denote the vertex of Cay(G, S) corresponding to the identity of G by 1.
A graph Γ is a Cayley graph on G if and only if AutΓ contains a subgroup which is regular on vertices and isomorphic to G. It is well-known that a Cayley graph is vertex-transitive. However, a Cayley graph is of course not necessarily arc-transitive. Thus much excellent work has dealt with arc-transitive Cayley graphs. In particular, there are many works about cubic and pentavalent Cayley graphs. For the cubic case, see [1][2][3] for cubic symmetric Cayley graphs on nite nonabelian simple groups, which are normal except for A , see [4] for a characterisation of connected cubic s-transitive Cayley graphs, see [5] for a classi cation of the connected arc-transitive cubic Cayley graphs on PSL( , p) where p is a prime, and see [6] for a classi cation of cubic arc-transitive Cayley graphs on a class of Frobenius groups. For the pentavalent case, see [7] for a classi cation of arc-transitive pentavalent Cayley graphs on nite nonabelian simple groups, see [8] for a construction of 2-arc transitive pentavalent Cayley graph of A , and see [9] for a characterization of connected core-free pentavalent 1-transitive Cayley graphs.
The objective of this paper is to give a characterization of pentavalent arc-transitive Cayley graphs on a class of primitive Frobenius groups with soluble vertex stabilizer.
A group G is said to be a Frobenius group if G has the form G = W:H such that xy ≠ yx for any x ∈ W \ { } and y ∈ H \ { }. In particular, G is called a primitive Frobenius group if H acts irreducibly on W.

Preliminary Results
In this section we give some preliminary results, which will be used in the subsequent sections. Let Γ be a G-vertex-transitive graph. Then, for α ∈ VΓ, the stabilizer Gα is a core-free subgroup in G, that is, ∩ g∈G G g α = .
Set H = Gα and D = {x α x ∈ Γ(α)}. Then D is a union of several double cosets HxH. Moreover, Γ is isomorphic to the coset graph Cos(G, H, D) de ned over {Hx x ∈ G} with edge set {{Hg , Hg } g g − ∈ D}.
The following statements for coset graphs are well known. The soluble vertex stabilizer for arc-transitive graphs of valency is known. We give two basic results for pentavalent graphs. Lemma 2.3. Let Γ = Cay(G, S) be a connected pentavalent graph with soluble stabilizer. Assume that AutΓ contains a subgroup X such that Γ is X-arc-transitive and G ¢ X. Then X ∼ = Z , D , or F .
Proof. Since Γ is connected, G = S , and thus Aut(G, S) acts faithfully on S. So Aut(G, S) S . By Lemma 2.1, X ≤ N AutΓ (G) = G:Aut(G, S). Thus X Aut(G, S) S . Note that X is transitive on S, so X ∼ = Z , D , or F .
We say a vertex-transitive graph Γ is a normal cover of its quotient graph Γ N if Γ and Γ N have the same valency, where N ¡ AutΓ is not transitive on VΓ.
Lemma 2.4. Let Γ be a connected pentavalent X-arc-transitive graph with soluble stabilizer, and let N ¡ X such that X/N is insoluble, where X ≤ AutΓ. Then Γ is a normal cover of Γ N .
Proof. Let u ∈ VΓ, and let B = u N be an orbit of N acting on VΓ. Let K be the kernel of X acting on VΓ N . Then Ku is soluble as Ku ¢Xu. By the Frattini argument, we have that K = NKu. Note that K/N ∼ = NKu /N ∼ = Ku /(N ∩Ku), so K/N is soluble. Since X/N is insoluble, X/K ∼ = (X/N)/(K/N) is insoluble. Thus Γ is a normal cover of Γ N .
The next lemma gives a classi cation of locally primitive Cayley graphs on abelian groups.

Examples
In this section we give some examples of pentavalent arc-transitive graphs.  Let F be the Fitting subgroup of A, that is, F is the largest nilpotent normal subgroup of A. Then F ̸ = , and C A (F) F as A is soluble.
For a group H and a prime p, we denote the Sylow p-subgroup of H by Hp.

Lemma 4.2. If G D p , then W is normal in A.
Proof. We claim that G ∩ F ≠ . Suppose that G ∩ F = . Since F ≤ Au, |F| |Au|. It follows that |F| by Lemma 2.2, where u ∈ VΓ.
Assume that F is transitive on VΓ. Then |G| |F|. Since H acts irreducibly on W, G ∼ = Z :Z or Z :Z . Since there exists no connected arc-transitive pentavalent graphs of order p for each prime p ≥ by [18,Theorem 1.1], the former case does not occur. By [17], there exists no connected arc-transitive pentavalent graphs of order , so the latter case is excluded. Similarly, we also exclude the case where Γ F ∼ = K .
Thus Γ is a normal cover of Γ F . Then |F| divides |G|. Since C G (F) F, C G (F) = . Therefore G acts faithfully on F. It follows that G Aut(F). Thus

|F|.
Suppose that p = . By the previous paragraph, we have that n . Assume that Φ(F) = . Since |F| divides both and |G|, we obtain that F ∼ = Z k and F ∼ = Z or , where k min{ , d}. Note that C A (F) F, so G GL(k, ). By Atlas [19], there exists no k and G satisfying the above relation. Thus Φ(F) ̸ = . Since , where k ≤ . By Atlas [19], F ∼ = Z and H ∼ = Z . It follows that |F | = and d = , which is a contradiction. Suppose that p is odd. Assume that Φ(F) = . Then F ∼ = Z k and k n, where k . Note that G GL(k, ), it follows that W GL(k, ), and Z k GL(k, ). By Atlas [19], there exists no k and W satisfying the above relation. [20, p.174, Theorem 1.4], a contradiction. Thus C W (F ) = , and so W GL(k, ), where k ≤ . By Atlas [19], this is impossible.
To sum up, F ∩ G ̸ = . Since W is minimal in G, W F. Note that G D p , so Γ is a cover of Γ Fp , and thus W = Fp. Therefore, W is normal in A. This completes the proof. Proof. By Lemma 4.2, W is normal in A. Since G D p , n > . Then Γ is a normal cover of Γ W . By Lemma 2.5, either G/W ¢ A/W or Γ W ∼ = K or K , . In the former case, G ¢ A. By Lemma 2.3, Au ∼ = Z , D , or F , where u ∈ VΓ. If Γ W ∼ = K , then A/W ≤ AutΓ W ∼ = S . Note that · |A/W|, so Au is insoluble, which is a contradiction. Similarly, we can exclude the case where Γ W ∼ = K , . This completes the proof of Lemma 4.3.
In the remaining section, we study the case where the full automorphism group A is insoluble. Denote by R the radical of A, that is, R is the largest soluble normal subgroup of A.
Suppose that R = . Then we have the following lemma. Proof. Let N be a minimal normal subgroup of A. Since R = , N = T × · · · × T ∼ = T , where T i ∼ = T is non-abelian simple. By [21], T is one of the following: Thus m . Suppose that T = PSL( , ). By Atlas [19], G i A and A has no subgroup with index , or . So m does not divide |Au|, which is a contradiction. Similarly, we can exclude the cases where T = PSU( , ), M , PSp( , ) and PSL( , q)(q < ).