Locally primitive Cayley graphs of dihedral groups

European Journal of Combinatorics 36 (2014) 39–52

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Locally primitive Cayley graphs of dihedral groups✩ Jiangmin Pan 1 School of Mathematics and Statistics, Yunnan University, Kunming, Yunnan 650091, PR China

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Article history: Received 9 March 2013 Accepted 9 June 2013 Available online 20 August 2013

abstract A graph Γ is called locally-primitive if the vertex stabilizer (AutΓ )α is primitive on the neighbor set of α for each vertex α . In this paper, we classify locally-primitive Cayley graphs of dihedral groups, while 2-arc-transitive Cayley graphs of dihedral groups have been classified by a series of papers in the literature. © 2013 Elsevier Ltd. All rights reserved.

1. Introduction Throughout the paper, by a graph Γ , we mean a connected, undirected and simple graph with valency at least three. For a graph Γ , denote its vertex set by V Γ . Let α ∈ V Γ and let Γ (α) = {β ∈ V Γ | β is adjacent to α} be the neighbor set of α . Then Γ is called X -locally-primitive, where X ≤ AutΓ , if the vertex stabilizer Xα := {x ∈ X | α x = α} acts primitively on Γ (α) for each vertex α . A graph Γ is called a Cayley graph of a group G if there is a subset S ⊆ G \{1}, with S = S −1 := {g −1 | g ∈ S }, such that V Γ = G, and two vertices g and h are adjacent if and only if hg −1 ∈ S. This Cayley graph is denoted by Cay(G, S ). For convenience, we sometimes call a Cayley graph of a dihedral group a dihedrant. It is well known that a graph Γ is isomorphic to a Cayley graph of a group G if and only if AutΓ contains a subgroup which is isomorphic to G and acts regularly on V Γ ; see [2, Proposition 16.3]. If this regular subgroup is normal in X with X ≤ AutΓ , then Γ is called an X -normal Cayley graph. In particular, if this regular subgroup is normal in AutΓ , then Γ is called a normal Cayley graph. Characterizing locally-primitive graphs is a current active topic in the area of algebraic graph theory; see for instance [5,10,11,23,21] and the references therein. The main purpose of this paper is to classify locally-primitive Cayley graphs of dihedral groups.

✩ This paper was partially supported by the National Natural Science Foundation of China and an ARC Discovery Grant Project.

E-mail address: [email protected]. 1 Tel.: +86 087167209493. 0195-6698/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ejc.2013.06.041

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For a graph Γ , a 2-arc of Γ is a sequence α, β, γ of distinct vertices such that α, γ ∈ Γ (β). A graph Γ is called 2-arc-transitive if AutΓ is transitive on the set of 2-arcs of Γ . A 2-arc-transitive graph is locally-primitive. For several classes of groups, their 2-arc-transitive Cayley graphs have been classified: see [16] for elementary abelian groups; see [1] for cyclic groups; see [20] for general abelian groups. In particular, 2-arc-transitive Cayley graphs of dihedral groups have been classified by a series of papers [24,25,7]. The proof of our classification of locally-primitive dihedrants depends on the classifications of primitive permutation groups which contain a cyclic regular subgroup obtained in [17], and those which contain a dihedral subgroup obtained in [19]. The terminology and notation used in this paper are standard. For example, for a positive integer n, we use Zn and D2n to denote the cyclic group of order n and the dihedral group of order 2n respectively; also, as in [6], we sometimes use n to denote a cyclic group of order n, use [n] to denote a group of order n, and use pn with p a prime to denote the elementary abelian group of order pn . Given two groups N and H, denote by N × H the direct product of N and H, by N · H an extension of N by H, and if such an extension is split, then we write N : H instead of N · H. There are some ‘trivial’ examples of 2-arc-transitive dihedrants: K2n , the complete graphs with 2n vertices; Kn,n , the complete bipartite graph of valency n; Kn,n − nK2 , the graph deleted a 1-match from Kn,n . Two sporadic examples arise from the Hadamard design on 11 points. For simplicity, we describe them in terms of an orbital graph. Recall that an orbital graph of a transitive permutation group X on a set Ω is a graph with vertex set Ω and arc set (α, β)X , where α, β ∈ Ω . Example 1.1. Let X = PGL(2, 11) and let H ∼ = A5 be a subgroup of X . Let Ω = [X : H ], of size 22. Then X acts on Ω bi-primitively, and has exactly two connected orbital graphs with valency 5 and 6. We denote these two graphs by H D (11, 5, 2) and H D (11, 6, 2) which are isomorphic to the incidence and non-incidence graphs of the Hadamard design on 11 points respectively. By Marušič [24, Theorem 1.2], both graphs are 2-arc-transitive dihedrants, and by Cheng and Oxley [4, Table 1], they both have the full automorphism group X . The point–hyperplane incidence and non-incidence graphs of projective geometries are also examples. Example 1.2. Let V (d, q) be a d-dimension linear space over q-elements field Fq , where d ≥ 3. Let Ω1 be the set of projective points (1-subspaces) of V (d, q), and let Ω2 be the set of hyperplanes ((d − 1)subspaces) of V (d, q). Define two graphs P H (d, q) and P H ′ (d, q): both have vertex set Ω1 ∪ Ω2 , the edge set of P H (d, q) is {{x, y} | x ∈ Ω1 , y ∈ Ω2 , x ⊂ y}, and the edge set of P H ′ (d, q) is d−1

{{x, y} | x ∈ Ω1 , y ∈ Ω2 , x ̸⊂ y}. Then the valency of P H (d, q) is q q−1−1 , the number of 1-subspaces of a (d − 1)-subspace, and the valency of P H ′ (d, q) equals |Ω1 | − val(P H (d, q)) = qd−1 . Both graphs are 2-arc-transitive dihedrants (see [24, Theorem 2.1]), and have the full automorphism group isomorphic to Aut(PSL(d, q)) = P Γ L(d, q) · Z2 (see [18, Theorem]). We remark that both graphs are (X , 2)-arc-transitive, where X ∼ = PGL(d, q) · Z2 is a subgroup of Aut(PSL(d, q)). Actually, for a vertex α , Xα ∼ = [qd−1 ] · GL(d − 1, q), and (AutΓ )α ∼ = Xα · ⟨σ ⟩, where Γ = P H (d, q) or P H ′ (d, q), and σ is the Frobenius automorphism. Then it is routine to show the 2-transitivity of XαΓ (α) by the 2-transitivity of (AutΓ )Γα (α) . A typical method for studying and constructing locally-primitive graphs is taking normal quotients. For a graph Γ , suppose X ≤ AutΓ has an intransitive normal subgroup N. Denote V ΓN the set of N-orbits in V Γ . The normal quotient graph ΓN of Γ induced by N is defined with vertex set V ΓN and two vertices B, C ∈ V ΓN are adjacent if and only if some vertex in B is adjacent in Γ to some vertex in C . If further Γ and ΓN have the same valency, then Γ is called a normal N-cover (or regular N-cover) of ΓN . The next family of examples, denoted by K2d q+1 , arises from a normal Zd -cover of the graph Kq+1,q+1 − (q + 1)K2 with q an odd prime power, which is first obtained in [7] and is a generalization of the graph K4d+1 obtained in [8]. Let Σ be a graph and K a finite group. A voltage assignment (or K -voltage assignment) of Σ is a function f : AΣ → K such that f (α, β) = f (β, α)−1 for each (α, β) ∈ AΣ , where AΣ denotes the

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arc set of Σ . Then the derived graph Σ ×f K from the voltage assignment f is defined with vertex set V Σ × K (Cartesian product), and (α, g ) is adjacent to (β, h) if and only if (α, β) ∈ AΣ and hg −1 = f (α, β). It is known that Σ ×f K is a normal cover of Σ , and each normal K -cover of Σ is isomorphic to Σ ×f K for some K -voltage assignment f of Σ . Example 1.3. Let q be an odd prime power and let Fq∗ = ⟨θ ⟩, where Fq∗ denotes the multiplicative group of the q-elements field Fq . Let d be a divisor of 2 if q ≡ 1 (mod 4), and a divisor of q − 1 if q ≡ 3 (mod 4). Let Σ = Kq+1,q+1 − (q + 1)K2 , with the vertex set two copies of the 1-dimensional projective geometry PG(1, q), and the missing matching in Σ consists of all pairs [i, i′ ], where i ∈ PG(1, q) and i′ is the other copy of i. ∼ Define the graph K2d q+1 as the normal Zd -cover Σ ×f K , where K = ⟨a⟩ = Zd and the voltage assignment f : AΣ → K is given as following: q−1

f (i, j′ ) =



1 ah

if i = ∞ or j = ∞; if i, j ̸= ∞, i − j = θ h .

The last examples arise from normal Cayley graphs of dihedral groups. Example 1.4. Let G = ⟨a, b | an = b2 = 1, ab = a−1 ⟩. Suppose r is an odd prime, and k is an integral solution of the congruence xr −1 + xr −2 + · · · + x + 1 ≡ 0 (mod n). Set



r −2 +kr −3 +···+1

S = {b, ab, ak+1 b, . . . , ak N D 2n,r ,k = Cay(G, S ).

b},

We will prove that the graphs N D 2n,r ,k with certain conditions are the only locally-primitive normal dihedrants (also the only locally-primitive normal Cayley graphs of metacyclic groups). The main result of this paper is the following theorem, which shows that the graphs introduced above are exactly all the locally primitive dihedrants. Theorem 1.5. Let Γ be a locally-primitive Cayley graph of a dihedral group of order 2n. Then one of the following statements is true, where q is a prime power. (1) Γ is 2-arc-transitive, and one of the following holds: (i) Γ = K2n , Kn,n or Kn,n − nK2 ; (ii) Γ = H D (11, 5, 2) or H D (11, 6, 2), the incidence or non-incidence graph of the Hadamard design on 11 points; (iii) Γ = P H (d, q) or P H ′ (d, q), the point–hyperplane incidence or non-incidence graph of (d−1)dimension projective geometry PG(d − 1, q), where d ≥ 3; q −1 if q ≡ 1 (mod 4), and a divisor of q − 1 if q ≡ 3 (mod 4), (iv) Γ = K2d q+1 , where d is a divisor of 2 respectively. e e (2) Γ = N D 2n,r ,k is a normal Cayley graph and is not 2-arc-transitive, where n = r t p11 p22 · · · pes s ≥ 13, with r , p1 , p2 , . . . , ps distinct odd primes, t ≤ 1, s ≥ 1 and r | (pi − 1) for each i. There are exactly (r − 1)s−1 non-isomorphic such graphs for a given order 2n. This paper is organized as follows. After this introduction section, we will give some preliminary results in Section 2. Then, by characterizing normal, vertex quasiprimitive and vertex biquasiprimitive cases in Section 3, and core-free case in Section 4, we complete the proof of Theorem 1.5 in Section 5. 2. Preliminary Let Γ = Cay(G, S ) be a Cayley graph of a group G. Let

ˆ = {ˆg | gˆ : x → xg , for all g , x ∈ G}, G Aut(G, S ) = {σ ∈ Aut(G) | S σ = S }.

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ˆ and Aut(G, S ) are subgroups of AutΓ . Further, the following nice property holds. Then both G ˆ) = Lemma 2.1 ([14, Lemma 2.1]). Let Γ = Cay(G, S ) be a Cayley graph. Then the normalizer NAutΓ (G ˆG : Aut(G, S ). ˆ is the right regular representation of G, and so is isomorphic (but not equal) to We remark that G ˆ as G. By Lemma 2.1, we easily have the following G. However, for convenience, we will often write G observation. Lemma 2.2. Let Γ = Cay(G, S ) be an X -normal locally-primitive Cayley graph of a group G, where G ▹ X ≤ AutΓ . Then Xα ≤ Aut(G, S ) and elements in S are involutions, where α denotes the vertex of Γ corresponding to the identity element of G. In particular, if G is abelian, then G is an elementary abelian 2-group. Giving a group X and two subgroups L and R such that L ∩ R is core-free in X (that is, L ∩ R contains no non-trivial normal subgroup of X ), define a bipartite graph (not necessarily connected) Cos(X , L, R), with vertex set [X : L] ∪ [X : R], and Lx and Ry are adjacent if and only if yx−1 ∈ RL. This graph is called a bi-coset graph. The following lemma is known. Lemma 2.3. Using notation as above. Let Γ = Cos(X , L, R). Then the following statements are true. (1) X ≤ AutΓ , and Γ is X -vertex-intransitive and X -edge-transitive. (2) Γ is connected if and only if ⟨L, R⟩ = X . (3) Γ is X -locally-primitive if and only if L ∩ R is a maximal subgroup of both L and R. Conversely, each X -vertex-intransitive and X -edge-transitive graph is isomorphic to Cos(X , Xα , Xβ ), where

α and β are adjacent vertices.

The following theorem provides a basic method for the studying of locally-primitive graphs (see [20, Lemma 2.5]), which slightly improves a remarkable result of Praeger [28, Theorem 4.1]. Theorem 2.4. Let Γ be an X -vertex-transitive locally-primitive graph, and let N ▹ X have at least three orbits on V Γ . Then the following statements hold. (1) N is semi-regular on V Γ , X /N ≤ AutΓN , ΓN is X /N-locally-primitive, and Γ is a normal N-cover of ΓN . (2) Γ is (X , s)-arc-transitive with s ≥ 2 if and only if ΓN is (X /N , s)-arc-transitive, where 1 ≤ s ≤ 5 or s = 7. (3) Xα ∼ = (X /N )δ , where α ∈ V Γ and δ ∈ V ΓN . Let G be an extension of N by H. If N is in the center of G, then the extension is called a central extension. A group G is said to be perfect if G = G′ , the commutator subgroup of G. For a given group H, if N is the largest abelian group such that G := N · H is perfect and the extension is a central extension, then N is called the Schur Multiplier of H. The following lemma is known. Lemma 2.5. Assume that G = N · T is a central extension with T a nonabelian simple group. Then G = NG′ and G′ = M · T , where M is a subgroup of the Schur Multiplier of T . We next give some technical lemmas. The first one is regarding complete bipartite graphs. Lemma 2.6. Let Γ = Kn,n , with bi-parts ∆1 and ∆2 , be an X -vertex-transitive locally-primitive graph, where X ≤ AutΓ . Let X + = X∆1 = X∆2 be the stabilizer of X acting on the bi-parts. Suppose that X + ∆

acts unfaithfully on ∆i with i = 1 or 2. Then Xα 3−i = (X + )∆3−i for each α ∈ ∆i . In particular, Γ is (X , 2)-arc-transitive if and only if X + is 2-transitive on ∆i .

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Proof. Let Ki be the kernel of X + acting on ∆i for i = 1, 2. Note that X = X + · Z2 and Xα = Xα+ . g g Choose g ∈ X \ X + . Then g 2 ∈ X + and X = ⟨X + , g ⟩. It is then easy to show that K1 = K2 , K2 = K1 and K1 K2 = K1 × K2 ▹ X . Let α ∈ ∆1 . Then Γ (α) = ∆2 . Since Γ = Kn,n is X -locally-primitive, ∆

∆

Xα 2 = XαΓ (α) is primitive, so is the overgroup (X + )∆2 . Then, as 1 ̸= K1 ∼ = K1 2 ▹ (X + )∆2 , we conclude that K1 is transitive on ∆2 . Similarly, K2 is transitive on ∆1 . Now, both X + and K2 act transitively on ∆1 , so |X + : Xα | = |X + : Xα+ | = |K2 : (K2 )α | = |∆1 |, it follows that |Xα : (K2 )α | = |X + : K2 |. Since Xα ∩ K2 is the kernel of Xα acting on ∆2 , we further have

|Xα∆2 | = |Xα : Xα ∩ K2 | = |Xα : (K2 )α | = |X + : K2 | = |(X + )∆2 |, ∆

∆

∆

hence Xα 2 = (X + )∆2 as Xα 2 ⊆ (X + )∆2 . Similarly, if α ∈ ∆2 , then Xα 1 = (X + )∆1 . Since Γ is (X , 2)-arc-transitive if and only if Γ is X -vertex-transitive and XαΓ (α) is 2-transitive, the last statement of the lemma is obviously true.  We remark that the automorphism groups which act locally-primitively or 2-arc-transitively on a complete bipartite graph have been systematically studied in [9,10], respectively. The following lemma follows directly from [21, Lemma 2.5]. Lemma 2.7. Let X = PGL(d, q) · o ≤ P Γ L(d, q) act naturally on the set of projective points of d-dimensional linear space over q-elements field, where o ≤ P Γ L(d, q)/PGL(d, q). If d ≥ 3, then X is not 2-primitive. A group X is called a central product of two subgroups M and N, denoted by X = M ∗ N, if X = MN, the commutator subgroup [M , N ] = 1, and M ∩ N is the center of X ; see [29, p. 141]. Moreover, a p-group X is called an extra-special p-group if Z (X ) = X ′ ∼ = Zp , where Z (X ) and X ′ denote the center and the commutator subgroup of X , respectively. Lemma 2.8. Let X be a group of order 32. Suppose the center Z (X ) ∼ = Z2 and X /Z (X ) ∼ = Z42 . Then either (1) X = D8 ∗ D8 , and Aut(X ) ∼ = Z42 · O+ 4 (2); or (2) X = D8 ∗ Q8 , and Aut(X ) ∼ = Z42 · O− 4 (2), where D8 and Q8 denote the dihedral group and the quaternion group of order 8 respectively, and O+ 4 (2) and O− ( 2 ) denote the orthogonal groups. 4 Proof. By assumption, X is an extra-special 2-group of order 32, then X = D8 ∗ D8 or D8 ∗ Q8 by Robinson [29, p. 141, Exercise 7]. Further, by Winter [30, Theorem 1], Aut(D8 ∗ D8 ) ∼ = Z42 · O+ 4 (2), and − 4 ∼ Aut(D8 ∗ Q8 ) = Z2 · O4 (2).  We give a property of special linear groups to end this section. Lemma 2.9. Let q be an odd prime power. Then SL(2, q) has no cyclic subgroup of order 2(q + 1), and has no dihedral subgroup. Proof. Since q is odd, SL(2, q) has a unique involution by Isaacs [15, Lemma 7.4], so SL(2, q) has no dihedral subgroup. If SL(2, q) has a cyclic subgroup H ∼ = Z2(q+1) , let Z denote the center of SL(2, q), then Z2 ∼ = Z ⊆ H and hence Zq+1 ∼ = H /Z ≤ SL(2, q)/Z = PSL(2, q), which is a contradiction as PSL(2, q) has no element of order q + 1.  3. Normal and basic locally primitive dihedrants A transitive permutation group X ≤ Sym(Ω ) is called quasiprimitive if each minimal normal subgroup of X is transitive on Ω , while X is called bi-quasiprimitive if each of its minimal normal subgroups has at most two orbits and there exists one which has exactly two orbits on Ω . By Theorem 2.4, it is easy to show that each vertex-transitive locally-primitive graph is a normal cover of a ‘basic graph’—vertex quasiprimitive or vertex bi-quasiprimitive locally-primitive graph. Generally, characterizing these basic graphs is an important step to work out the general ones.

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In this section, we will determine normal locally-primitive dihedrants, and give some characterizations of basic locally-primitive dihedrants. Recall a graph Γ is called one-regular (or arcregular) if AutΓ is regular on the arc set of Γ . Lemma 3.1. Let Γ be an X -normal locally-primitive Cayley graph of a dihedral group of order 2n, where e e X ≤ AutΓ . Then Γ ∼ = N D 2n,r ,k as in Example 1.4, with n = r t p11 p22 · · · pes s , where r , p1 , p2 , . . . , ps are distinct odd primes, t ≤ 1, s ≥ 0 and r | (pi − 1) for each i. Further, Γ is a locally-primitive normal Cayley graph of a dihedral group of order 2n if and only if Γ ∼ = N D 2n,r ,k with n further satisfying n ≥ 13 and s ≥ 1. There are exactly (r − 1)s−1 non-isomorphic graphs N D 2n,r ,k of order 2n. Proof. Suppose G = ⟨a, b | an = b2 = 1, ab = a−1 ⟩ ▹ X and Γ = Cay(G, S ). By Lemma 2.2, elements in S are involutions and Xα ≤ Aut(G, S ) ≤ Aut(G) ∼ = Zn : Z∗n , where α denotes the vertex of Γ ∗ corresponding to the identity element of G, and Zn denotes the group of units of the ring Zn . Since Γ is connected, ⟨S ⟩ = G and so Xα acts faithfully on Γ (α) = S. Then, because Γ is X -locally-primitive, Xα ∼ = XαS is primitive, it then follows from Xα ≤ Zn : Z∗n that Xα ∼ = Zr : Zh for some odd prime r and a divisor h of r − 1. In particular, val(Γ ) = r and S = s⟨σ ⟩ , where s ∈ S is an involution, and σ ∈ Aut(G, S ) is of order r. If s ∈ ⟨a⟩, then ⟨S ⟩ ⊆ ⟨a⟩ ̸= G, which contradicts the connectivity of Γ . So s ∈ G \ ⟨a⟩. Since Cay(G, S ) ∼ = Cay(G, S φ ) for each φ ∈ Aut(G), and involutions in G \ ⟨a⟩ are conjugate to b in Aut(G), without loss of generality, we may suppose s = b. It is known that σ (actually each automorphism of G) has a form

σ : a → ak ,

b → al b ,

where (k, n) = 1, 1 ≤ l ≤ n.

By direct computation, S = b⟨σ ⟩ = {b, al b, . . . , a(k +k +···+1)l b}. Because ⟨S ⟩ = G and o(σ ) = r, we easily conclude that (l, n) = 1 and kr −1 + kr −2 + · · · + 1 ≡ 0 (mod n). It then follows e e from [12, Lemma 3.3] that n = r t p11 p22 · · · pes s , where r , p1 , p2 , . . . , ps are distinct odd primes, t ≤ 1 and r | (pi − 1) for each i. Now, let τ ∈ Aut(G) such that τ : al → a, b → b. Then r −2

r −3

S τ = {b, ab, ak+1 b, . . . , ak +k +···+1 b}. Hence Γ = Cay(G, S ) ∼ = Cay(G, S τ ) = N D 2n,r ,k , as in Example 1.4. Obviously, ⟨σ ⟩ ⊆ Aut(G, S ) ≤ AutΓ , so G : ⟨σ ⟩ ≤ AutΓ acts regularly on the arc set of Γ . If n ≥ 13 and s ≥ 1, by Feng and Li [12, Theorem 3.1], N D 2n,r ,k is one-regular, so AutΓ = G : ⟨σ ⟩, that is, Γ ∼ = N D 2n,r ,k is a normal Cayley graph of G. Again by Feng and Li [12, Theorem 3.1], there are exactly (r − 1)s−1 non-isomorphic graphs N D 2n,r ,k for a given n. On the other hand, if s = 0, then n = 2r and Γ = Kr ,r is not a normal Cayley graph as AutΓ = Sr ≀ Z2 has no normal subgroup of order 2r; if s ≥ 1 and n < 13, by the restriction of n, the only possibilities are (n, r ) = (7, 3) and (11, 5). Edge-transitive graphs of order twice a prime are classified in Table 1 of [4]. Checking the Table, for (n, r ) = (7, 3), Γ = N D 14,3,k ∼ = P H (3, 2) and AutΓ ∼ = P Γ L(3, 2) : Z2 , which has no normal subgroup of order |V Γ | = 14; and for (n, r ) = (11, 5), Γ = N D 22,5,k ∼ = H D (11, 5, 2) and AutΓ ∼ = PGL(2, 11) has no normal subgroup of order |V Γ | = 22, both graphs are not normal Cayley graphs.  r −2

r −3

We remark that the locally-primitive normal dihedrants N D 2n,r ,k in Lemma 3.1 are also the only locally-primitive normal Cayley graphs of metacyclic groups, as it is showed in [26, Theorem 1.1] that locally-primitive normal Cayley graphs of metacyclic groups are normal dihedrants. A transitive permutation group X on Ω is called 2-primitive if the stabilizer Xα with α ∈ Ω is primitive on Ω \ {α}. Obviously, a 2-primitive permutation group is 2-transitive, and a 3-transitive permutation group is 2-primitive. For convenience, we sometimes call a permutation group a c-group or d-group if it has a regular subgroup which is cyclic or dihedral, respectively. The quasiprimitive c-groups are determined in [17], stated as following. For the discussion of the subsequent sections, the transitivity of the groups are also given. Theorem 3.2. Let X ≤ Sym(Ω ) be a quasiprimitive permutation group which contains a regular cyclic subgroup G ∼ = Zn . Then the triple (X , n, Xα ) is listed in Table 1, where α ∈ Ω , q is a prime power, o ≤ P Γ L(d, q)/PGL(d, q) and P1 ∼ = [qd−1 ] : GL(d − 1, q). In particular, if X is almost simple, then X is 2-primitive if and only if X ̸= PGL(d, q) · o with d ≥ 3.

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Table 1 Quasiprimitive c-groups. X

n

Xα

Conditions

Transitivity

Zp : Zl An Sn

p n n 11 23 11

Zl An−1 Sn−1

l | (p − 1) n ≥ 5 odd n≥4

2-transitive iff l = p − 1 (n − 2)-transitive n-transitive 4-transitive 4-transitive 2-primitive

d≥2

2-transitive if d ≥ 3 3-transitive if d = 2

M11 M23 PSL(2, 11)

PGL(d, q) · o

M10 M22 A5

qd −1 q−1

P1 · o

Table 2 Quasiprimitive d-groups. X

G

Xα

A4 S4 AGL(3, 2) AGL(4, 2) 24 : A7 24 : S6 24 : A6 24 : S5 24 : Γ L(2, 4)

D4 D4 D8 D16 D16 D16 D16 D16 D16

Z3 S3 GL(3, 2) GL(4, 2) A7 S6 A6 S5 Γ L(2, 4)

2-primitive 4-transitive 3-transitive 3-transitive 3-transitive 2-primitive 2-primitive 2-transitive 2-transitive

M12 M22 · 2 M24 S2m A4m PSL(2, pe ) · o

D12 D22 D24 D2m D4m Dpe +1

M11 PSL(3, 4) · 2 M23 S2m−1 A4m−1

PGL(2, pe ) · Zf

Dp e + 1

Conditions

Transitivity

Zep : Z pe −1 · o

pe ≡ 3 (mod 4), o ≤ Z2 × Ze

5-transitive 3-transitive 5-transitive 2m-transitive (4m − 2)-transitive 3-transitive iff

Zep : Zpe −1

pe ≡ 1 (mod 4), f | e

o ≥ Z2 is not Frobenius 3-transitive

2

The quasiprimitive d-groups are classified in [19, Theorem 1.5], with two examples missed and pointed out on [22], namely, the affine groups AΓ L(2, 4) ∼ = Z42 : Γ L(2, 4) ∼ = Z42 : (Z3 × A5 ) · Z2 4 ∼ and AΣ L(2, 4) = Z2 : S5 , both containing a regular dihedral subgroup D16 . The following is a correct version. Theorem 3.3. Let X ≤ Sym(Ω ) be a quasiprimitive permutation group which contains a regular dihedral subgroup G. Then X is 2-transitive, and the triple (X , G, Xα ) is listed in Table 2, where α ∈ Ω . Theorem 3.3 has an immediate consequence regarding dihedrants. Corollary 3.4. A vertex quasiprimitive Cayley graph Γ of a dihedral group is a vertex 2-transitive complete graph. Further, Γ is X -locally-primitive with X ≤ AutΓ if and only if X is 2-primitive on V Γ , and Γ is (X , 2)-arc-transitive if and only if X is s-transitive on V Γ with s ≥ 3. Let X be a bi-quasiprimitive permutation group on Ω . Let N be a non-trivial intransitive normal subgroup of X . Then N has exactly two orbits on Ω , say ∆1 and ∆2 . Obviously, {∆1 , ∆2 } forms an X invariant partition on Ω . This partition is called a normal partition. A simple lemma in [22] shows that if |Ω | > 4, then X has a unique normal partition. A transitive permutation group X on Ω is called bi-primitive if Ω has a non-trivial X -invariant partition {U , V } such that XU = XV acts primitively on both U and V . The next two lemmas show that bi-quasiprimitive c-groups and bi-quasiprimitive d-groups are bi-primitive. Lemma 3.5 ([20]). Let X ≤ Sym(Ω ) be a bi-quasiprimitive permutation group containing a regular cyclic subgroup, with a normal partition {∆1 , ∆2 }. Then X is bi-primitive, and X∆1 = X∆2 is primitive on ∆1 and ∆2 .

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Lemma 3.6 ([27, Theorem 2.4]). Let X ≤ Sym(Ω ) be a bi-quasiprimitive permutation group which contains a regular dihedral group G. Let {∆1 , ∆2 } be a normal partition of X on Ω . Then X is bi-primitive, X + := X∆1 is primitive on ∆1 and ∆2 , and one of the following is true, where α ∈ Ω . acts faithfully on ∆i , and one of the following holds: X = Zp : Z2l ≤ AGL(1, p), G = D2p and Xα ∼ = Zl , where p is a prime and 2l | (p − 1); (X , G, Xα ) = (S4 , D8 , Z3 ), (PGL(2, 11), D22 , A5 ) or (M12 · 2, D24 , M11 ); X = PGL(2, q) · Zf , G = D2(q+1) and Xα = Zep : Z(q−1)/(2,q−1) · Zf , where q = pe ≡ 3 (mod 4) and f | e; (d) X = PGL(d, q) · o · 2 ≤ Aut(PSL(d, q)), G = D2(qd −1)/(q−1) and Xα = [qd−1 ] · GL(d − 1, q) · o, where d ≥ 3 and o ≤ P Γ L(d, q)/PGL(d, q); (e) (X , G, Xα ) = (S4m , D8m , A4m−1 ) or (S2m+1 , D4m+2 , A2m ). (2) X + acts unfaithfully on ∆i , K1 × K2 ▹ X + ≤ (X + )∆1 × (X + )∆2 = K1 · o1 × K2 · o2 , where Ki is the kernel of X + acting on ∆i with i = 1, 2, and Ki · oi is a primitive c-group or a primitive d-group on ∆3−i , as listed in Tables 1 or 2 respectively. (1) X + (a) (b) (c)

The following lemma determines the automorphism groups that act locally-primitively but not 2-arc-transitively on vertex bi-quasiprimitive dihedrants, which will be used later. Lemma 3.7. Let Γ = Cay(G, S ) be an X -locally-primitive Cayley graph of a dihedral group G, where G ≤ X ≤ AutΓ . Suppose further X is bi-quasiprimitive on V Γ . Then Γ is (X , 2)-arc-transitive with the only exceptions as following, where α ∈ V Γ . (i) Γ = N D 2p,r ,k , X = Zp : Z2r ≤ AGL(1, p), G = D2p and Xα = Zr , where r is an odd prime divisor of p − 1. (ii) Γ = K4,4 − 4K2 , and (X , G, Xα ) = (S4 , D8 , Z3 ). (iii) Γ = Kq+1,q+1 − (q + 1)K2 , X = PGL(2, q) · Zf , G = D2(q+1) , and Xα = Zep : Z(q−1)/(2,q−1) · Zf , where q = pe ≡ 3 (mod 4) and f | e. (iv) Γ = Kp,p , G = D2p and X = ((Zp : Zr ) × (Zp : Zr )) · Zs · Z2 ≤ AGL(1, p) ≀ Z2 , where rs is a proper divisor of p − 1. Proof. Let {∆1 , ∆2 } be a normal partition of X on V Γ . Let X + = X∆1 = X∆2 . Then X = X + · 2, and X + is primitive on ∆1 and ∆2 by Lemma 3.6. Let α ∈ ∆1 . Since Γ is X -locally-primitive, Xα = Xα+ is primitive on Γ (α). We divided our proof into the following two cases. (1) Suppose that X + ∼ = (X + )∆i acts faithfully on ∆i . Then X satisfies Lemma 3.6(1). If X satisfies part (a) of Lemma 3.6(1), then X = Zp : Z2l with 2l | (p − 1), G = D2p and Xα = Zl . Since Xα is primitive on Γ (α), l is an odd prime and val(Γ ) = l. Set r = l. Noting that G ▹ X , Γ is an X -normal locally-primitive dihedrants of D2p . By Lemma 3.1, Γ = N D 2p,r ,k , as in part (i). If (X , G, Xα ) = (S4 , D8 , Z3 ), then val(Γ ) = 3, |V Γ | = 8, and Γ = K4,4 − 4K2 is not (X , 2)-arctransitive, as in part (ii). Consider the other candidates listed in Lemma 3.6(1). Then X + is always almost simple and 2-transitive on ∆i . Assume first (X + )∆1 and (X + )∆2 are permutation inequivalent. Then X + has at least two faithful 2-transitive permutation representations. By Cameron [3, Notes 4], we have that either (X , G, Xα ) = (PGL(2, 11), D22 , A5 ) or (X , G, Xα ) satisfies part (d) of Lemma 3.6(1). For the former case, by Example 1.1, Γ = H D (11, 5, 2) or H D (11, 6, 2) is (X , 2)-arc-transitive. For the latter case, ∆1 and ∆2 can be identified with the set of projective points and the set of hyperplanes, respectively. If there is a hyperplane γ ∈ Γ (α) such that α is contained in γ , then Γ = P H (d, q), and otherwise Γ = P H ′ (d, q). Since X ≥ PGL(d, q) · 2, both graphs are (X , 2)-arc-transitive by the remark in Example 1.2. Assume next that (X + )∆1 and (X + )∆2 are permutation equivalent. We claim that (X , G, Xα ) ̸= (PGL(2, 11), D22 , A5 ) and (M12 · 2, D24 , M11 ). If not, let g ∈ X \ X + and β = α g ∈ ∆2 , since (X + )∆1 and (X + )∆2 are permutation equivalent, Xβ = Xα g = Xαg is conjugate to Xα in X + , that is, Xβ = Xαh −1

for some h ∈ X + . It follows that Xαgh = Xα and so gh−1 ∈ NX (Xα ); noting that, for the two cases, Xα are self-normalized in X , we further conclude that g ∈ Xα h ⊆ X + , which is a contradiction. Thus the

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claim is true, and hence X satisfies parts (c)–(e) of Lemma 3.6(1). Further, as (X + )∆1 and (X + )∆2 are permutation equivalent, Xα fixes a vertex γ ∈ ∆2 and is transitive on ∆2 \ {γ }. So Γ = Kn,n − nK2 , where n = |∆i |. It is easy to see that X acts locally-primitively but not 2-arc-transitively on Γ = Kn,n − nK2 if and only if X + is 2-primitive but not 3-transitive on ∆i . Assume that X satisfies part (c) or (d) of Lemma 3.6(1). Then ∆i may be identified with the set of projective points. For part (c), |∆i | = q + 1, since pe ≡ 3 (mod 4), e is odd, so is f . It follows that X + = PSL(2, q) · Zf is 2-primitive but not 3-transitive on ∆i , which gives rise to examples, as in part (iii); for part (d), since d ≥ 3, by Lemma 2.7, X + = PGL(d, q) · o ≤ P Γ L(d, q) is not 2-primitive on ∆i , which is a contradiction. Assume that X satisfies part (e) of Lemma 3.6(1). Then |∆i | = 4m or 2m + 1, and X + = A4m or A2m+1 is s-transitive on ∆i with s ≥ 3, respectively. Hence Γ is (X , 2)-arc-transitive. (2) Suppose that X + acts unfaithfully on ∆i . Let Ki be the kernel of X + acting on ∆i , where ∆ i = 1, 2. Then X + and Ki satisfy Lemma 3.6(2). Since 1 ̸= K1 ∼ = K1 2 ▹ (X + )∆2 and (X + )∆i is primitive, we have that Xα ⊇ K1 is transitive on ∆2 . Thus Γ = Kn,n . By Lemma 2.6, Γ is (X , 2)arc-transitive if and only if (X + )∆i is 2-transitive. Noting that G ∩ X + is either a cyclic group or a dihedral group and is regular on ∆i , (X + )∆i is a primitive c-group or a primitive d-group. Checking the candidates in Theorems 3.2–3.3, we conclude that either (X + )∆i is 2-transitive, or |∆i | = p is an odd prime and (X + )∆i = Zp : Zl ≤ AGL(1, p), where l is a proper divisor of p − 1. The former case corresponds (X , 2)-arc-transitive graphs Γ . For the latter case, Γ = Kp,p is X -locally-primitive but not (X , 2)-arc-transitive. Moreover, as K1 ▹ (X + )∆2 , K1 ∼ = Zp : Zr for some r | l; as K1 ∼ = K2 and K1 × K2 ▹ X + , X + = (K1 × K2 ) · P ∼ = ((Zp : Zr ) × (Zp : Zr )) · P for some group P. Then, because (Zp : Zr ) · P ∼ = X + /K2 ∼ = (X + )∆2 ∼ = Zp : Zl , we have P ∼ = Zs with rs = l. Hence + X = X · 2 = ((Zp : Zr ) × (Zp : Zr )) · Zs · 2 ≤ AGL(1, p) ≀ Z2 , as in part (iv).  4. Locally-primitive core-free dihedrants Suppose Γ = Cay(G, S ) and G ≤ X ≤ AutΓ . If G is core-free in X , then Γ is called a core-free Cayley graph of G (noting that G is core-free in X imply that G is core-free in AutΓ ). In this section, we characterize locally-primitive core-free dihedrants, which will play a crucial role in the proof of Theorem 1.5. Recall that, for a group B, the socle of B, denoted by soc(B), is the product of all minimal normal subgroups of B. Lemma 4.1. Let Γ be an X -locally-primitive core-free Cayley graph of a dihedral group G, where G ≤ X ≤ AutΓ . Suppose further Γ is not (X , 2)-arc-transitive. Then either (1) X is 2-primitive but not 3-transitive on V Γ , and Γ is a complete graph; or (2) X is bi-quasiprimitive on V Γ , and X and Γ satisfy parts (iii)–(iv) of Lemma 3.7. In particular, Γ is 2-arc-transitive, and X < AutΓ . Proof. If X is quasiprimitive on V Γ , then part (1) follows from Corollary 3.4. If X is bi-quasiprimitive on V Γ , by Lemma 3.7, part (2) holds by noticing that G is not core-free in X in parts (i)–(ii) of Lemma 3.7. Thus, assume that X is neither quasiprimitive nor bi-quasiprimitive on V Γ in the following. We will prove that no example appears in this case. By assumption, X has a normal subgroup N which is a maximal subject that has at least three orbits on V Γ . By Theorem 2.4, N is semi-regular on V Γ , and X¯ := X /N ≤ AutΓN . Let α ∈ V Γ and δ ∈ V ΓN . Let G = ⟨a, b | an = b2 = 1, ab = a−1 ⟩ and ¯ = GN /N. Since G ∩ N ▹ G and |G : G ∩ N | ≥ |G|/|N | ≥ 3, we conclude that G ∩ N < ⟨a⟩ is a G ¯ ∼ cyclic group and G = G/(G ∩ N ) is a dihedrant group. Set G¯ = ⟨¯a⟩ : ⟨b¯ ⟩. Since G¯ ≤ X¯ is a transitive ¯ δ is core-free in G, ¯ so either G¯ δ = 1 or G¯ δ = ⟨¯ai b¯ ⟩ ∼ permutation group on V ΓN , G = Z2 for some i. For the former case,

|G| |G| |G| = N = |V ΓN | = |G¯ | = , |N | |α | |G ∩ N | so N ≤ G, which is not possible as G is core-free in X .

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¯δ ∼ Consider the latter case. Then G = Z2 and H := ⟨¯a⟩ ≤ G¯ is regular on V ΓN , so G¯ and X¯ are c-groups. Moreover, as |G| 1 1 |G| |G| = N = |V ΓN | = |G¯ | = , |N | |α | 2 2 |G ∩ N | we have |N | = 2|G ∩ N |, so N = (G ∩ N ) · 2. As G ∩ N is cyclic, N is a cyclic group or a metacyclic group. If G ∩ N is not a 2-group, then Hall 2′ -subgroup N2′ = (G ∩ N )2′ ⊆ G is a non-trivial normal subgroup of X , which is a contradiction. Thus G ∩ N and so N are 2-groups. Assume |G ∩ N | ≥ 4. Let Φ (N ) be the Frattini subgroup of N (that is, Φ (N ) is the intersection of all maximal subgroups of N). Since N is a cyclic or metacyclic 2-group, N /Φ (N ) ≤ Z22 , it follows that Φ (N ) ≤ G ∩ N is a non-trivial normal subgroup of X , yielding a contradiction. So G ∩ N ≤ Z2 and |N | divides 4. If N ∼ = Z4 , then Z2 ∼ = G ∩ N ▹ X , which contradicts that G is core-free in X . Hence N ∼ = Z2 or Z22 . Moreover, by the maximality of N, X¯ is a quasiprimitive or bi-quasiprimitive c-group on V ΓN , and by Theorem 2.4, ΓN is X¯ -locally-primitive but not (X¯ , 2)-arc-transitive. Case 1. Assume that X¯ is a quasiprimitive c-group on V ΓN . Then the triple (X¯ , |V ΓN |, X¯ δ ) is listed (as (X , n, Xα ) there) in Table 1 of Theorem 3.2. If X¯ is an affine group, then X¯ = Zp : Zl and X¯ δ ∼ = Zl , or X¯ = S4 and X¯ δ ∼ = S3 . For the former, since X¯ δ is primitive on ΓN (δ), l is an odd prime. So ΓN is of odd order p and odd valency l, which is impossible. For the later, ΓN ∼ = K4 is (X¯ , 2)-arc-transitive, which is a contradiction. Thus, X¯ is almost simple. By Theorem 3.2, X¯ is 2-transitive on V ΓN , so ΓN is a complete graph. It follows that X¯ is 2-primitive but not 3-transitive on V ΓN . By Theorem 3.2, the only possibility is X¯ = PSL(2, 11) and |V ΓN | = 11. Since N ∼ = Z2 or Z22 , X = N · PSL(2, 11) is a central extension. By ′ Lemma 2.5, X = NX , and since the Schur Multiplier of PSL(2, 11) is Z2 , we have X ′ = PSL(2, 11) or SL(2, 11). It follows that there are two possibilities as following: (i) N ∼ = Z2 , G (ii) N ∼ = Z22 , G

∼ = D22 , and X ∼ = D44 , and X

∼ = Z2 × PSL(2, 11) or SL(2, 11); ∼ = Z22 × PSL(2, 11) or ⟨c ⟩ · (Z2 × PSL(2, 11)), where ⟨c ⟩ = N ∩ X ′ ∼ = Z2 .

For case (i), obviously, Z2 × PSL(2, 11) has no subgroup isomorphic to D22 , and by Lemma 2.9, so is SL(2, 11), a contradiction occurs. For case (ii), if X ∼ = Z2 is a normal subgroup of X , which contradicts = Z22 × PSL(2, 11), then G ∩ N ∼ that G is core-free in X ; if X ∼ = ⟨c ⟩ · (Z2 × PSL(2, 11)), then ⟨c ⟩ is not contained in G as ⟨c ⟩ ▹ X , so D44 ∼ =G∼ = G/(G ∩ ⟨c ⟩) ∼ = G⟨c ⟩/⟨c ⟩ ≤ X /⟨c ⟩ ∼ = Z2 × PSL(2, 11), which is not possible. Case 2. Assume that X¯ is a bi-quasiprimitive c-group on V ΓN . Then ΓN is a bipartite graph, with a normal partition {∆1 , ∆2 } say. Let X¯ + = X¯ ∆1 = X¯ ∆2 . Then X¯ = X¯ + · Z2 , H ∩ X¯ + is a regular cyclic group on ∆i , and X¯ + acts primitively on ∆i by Lemma 3.5. Let Ki be the kernel of X¯ + acting on ∆i , where i = 1 or 2. Choose an element g ∈ X¯ \ X¯ + . Then g g 2 g ∈ X¯ + , and it is easy to show that K1 = K2 , K2 = K1 , and K1 × K2 ▹ X¯ .

Suppose first K1 = 1. Then X¯ + is a primitive c-group on both ∆1 and ∆2 . Hence the triple (X¯ + , |∆i |, X¯ δ+ ) is listed (as (X , n, Xα ) there) in Table 1 of Theorem 3.2. If X¯ + is an affine group, then either (X¯ + , |∆i |, X¯ δ+ ) = (S4 , 4, S3 ), or X¯ + = Zp : Zl and X¯ δ+ ∼ = Zl with p a prime. For the former case, ΓN = K4,4 − 4K2 is (X¯ , 2)-arc-transitive, a contradiction appears. For the latter case, as X¯ δ is primitive on ΓN (δ), l is an odd prime, which is not possible as Z2 ∼ = G¯ δ ≤ X¯ δ ∼ = Zl . Assume that X¯ + is almost simple. Then X¯ + ∼ = (X¯ + )∆i is 2-transitive by Theorem 3.2. Suppose (X¯ + )∆1 and (X¯ + )∆2 are permutation equivalent. Let δ ∈ ∆1 . Then X¯ δ+ has a fixed vertex δ ′ ∈ ∆2 , and has two orbits ∆1 \ {δ} and ∆2 \ {δ ′ }. Thus ΓN (δ) = ∆2 \ {δ ′ } and ΓN = Km,m − mK2 , where m = |∆i |. If (X¯ + , X¯ δ+ ) = (PSL(2, 11), A5 ), then X¯ = X¯ + · 2 ∼ = PGL(2, 11) by the bi-quasiprimitivity of X¯ . Since |V ΓN | = 22 and G¯ δ = Z2 , we have G¯ ∼ D , which is not possible as X¯ = PGL(2, 11) has no subgroup = 44 isomorphic to D44 . If soc(X¯ + ) = PSL(d, q) with d ≥ 3, then X¯ + is not 2-primitive on V ΓN by Lemma 2.7, so X¯ δ = X¯ δ+ is not primitive on ΓN (δ) = ∆2 \ {δ ′ }, that is, ΓN is not X¯ -locally-primitive, which is a contradiction.

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For the remaining cases, by Theorem 3.2, X¯ + is always 3-transitive on ∆i . It follows that X¯ δ is 2-transitive on ΓN (δ) = ∆2 \ {δ ′ }. Thus ΓN is (X¯ , 2)-arc-transitive, also yielding a contradiction. Suppose (X¯ + )∆1 and (X¯ + )∆2 are permutation inequivalent. Then X¯ + has at least two faithful 2-transitive permutation representations of degree |∆i |. Checking the candidates in Theorem 3.2 and by Cameron [3, Theorem 5.3], we conclude that either X¯ + = PSL(2, 11) and X¯ δ = A5 , or PGL(d, q) ▹ X¯ + with d ≥ 3. It follows that X¯ = PGL(2, 11) or X¯ ≥ PGL(d, q) · 2 respectively. Then, as discussed in Lemma 3.7, we obtain that ΓN = H D (11, 5, 2) or H D (11, 6, 2) for the former case, and ΓN = P H (d, q) or P H ′ (d, q) for the latter case. The four graphs are (X¯ , 2)-arc-transitive by Examples 1.1–1.2, which is a contradiction. Now, suppose K1 ̸= 1. Since (X + )∆2 is primitive c-group and 1 ̸= K1 ∼ = (K1 )∆2 ▹ (X¯ + )∆2 , we have that Xα ⊇ K1 is transitive on ∆2 . Hence Γ = Km,m with m = |∆i |. By Theorem 3.2, (X¯ + )∆i is either 2-transitive, or (X¯ + )∆i = Zp : Zl ≤ AGL(1, p) with l a proper divisor of p − 1. Then Lemma 2.6 implies that ΓN is (X¯ , 2)-arc-transitive for the former case, which is a contradiction. Consider the latter case. If p = 3, then l = 1 and Ki ∼ = (X¯ + )∆i ∼ = Z3 , so |X¯ | = 2 · 32 = 18; however, as G¯ δ ∼ = Z2 and |V ΓN | = 6, we ¯ | = 12, which is a contradiction as G¯ ⊆ X¯ . Suppose p ≥ 5. Let X¯ p be a Sylow p-subgroup of X¯ . have |G Then X¯ p ∼ = Z2p has exactly two orbits on V ΓN . Since N ∼ = Z2 or Z22 , we have that Y := N · X¯ p = N × Xp ∼ has exactly two orbits on V Γ , and Yα = Zp is regular on Γ (α). Hence Γ is a Y -vertex intransitive locally-primitive graph. By Lemma 2.3, Γ = Cos(Y , Yα , Yβ ) for some β ∈ Γ (α). However, as Y is abelian and Yα ∼ = Yβ ∼ = Zp , ⟨Yα , Yβ ⟩ ̸= Y , Γ is not connected, which is also a contradiction. Finally, since the graphs in parts (iii)–(iv) of Lemma 3.7 are 2-arc-transitive, the last statement of Lemma 4.1 follows.  5. Proof of Theorem 1.5 We will prove our main result (Theorem 1.5) in this section. Let Γ be a locally-primitive Cayley graph of a dihedral group G. Suppose G = ⟨a, b | an = b2 = 1, ab = a−1 ⟩ ∼ = D2n . Let A = AutΓ and let K = coreA (G), the largest normal subgroup of A contained in G. Then K is semi-regular on V Γ , and G/K is core-free in A/K . The case K = G has been characterized in Lemma 3.1. The next lemma treats the case where |G : K | = 2. Lemma 5.1. Suppose |G : K | = 2. Then Γ = K4,4 or K4,4 − 4K2 . Proof. Since |G : K | = 2, G is not normal in A, Γ is a bipartite graph, and K ∼ = Zn , or K ∼ = Dn with n even. Assume n = 2m is even. If m = 2, then |V Γ | = 8, so Γ = K4,4 or K4,4 − 4K2 . Also, it is easy to check that both graphs are really examples. Suppose m ≥ 3. Then ⟨a2 ⟩ ∼ = Zm is a normal subgroup of A and has exactly four orbits on V Γ . By Theorem 2.4, the normal quotient graph Γ⟨a2 ⟩ is an A/⟨a2 ⟩-locally-primitive graph of order 4. So Γ⟨a2 ⟩ ∼ = K4 and A/⟨a2 ⟩ ≤ Aut(K4 ) = S4 . Since

Z22 ∼ = G/⟨a2 ⟩ ≤ A/⟨a2 ⟩ ≤ S4 , we have G/⟨a2 ⟩ ▹ A/⟨a2 ⟩, implying that G ▹ A, which is a contradiction. Assume now n is odd. Then K ∼ = Zn has two orbits on V Γ , say ∆1 and ∆2 . Let A+ = A∆1 and α ∈ ∆1 . By Giudici et al. [13, Lemma 5.2], either Γ = Kn,n or A+ is faithful on both ∆1 and ∆2 . For the former case, A = AutΓ = Sn ≀ Z2 has no normal cyclic subgroup of order n, which contradicts that Zn ∼ = K ▹ A. For the latter case, as K is regular on ∆1 and A+ = K : Aα acts faithfully on ∆1 , we obtain ∼ + that the centralizers CAα (K ) = 1 and CA+ (K ) = K . Hence Aα = A+ α = A /K ≤ Aut(K ) is abelian. Γ (α) Γ (α) ∼ Now, Aα is an abelian primitive permutation group, so Aα = Zr is regular and val(Γ ) = r for some odd prime r. Let β ∈ Γ (α). Then Aαβ fixes Γ (α) and Γ (β) pointwise. By the connectivity of Γ , we have Aαβ = 1. So Aα acts faithfully on Γ (α), and Aα ∼ = AΓα (α) ∼ = Zr is regular on Γ (α). That is, Γ is a one-regular Cayley graph of the dihedral group G and of odd prime valency r. By Feng and Li [12, Theorem 3.1], Γ ∼ = N D 2n,r ,k is a normal Cayley graph of G, which is also a contradiction.  Thus, suppose that K ̸= 1 has at least three orbits on V Γ in the following. As |G : K | ≥ 3, K < ⟨a⟩ is ¯ = G/K . By Theorem 2.4, a cyclic group, and G/K is a regular dihedral group on V ΓK . Let A¯ = A/K and G ¯ ¯ We remark here, although ΓK is an A-locally-primitive core-free Cayley graph of the dihedral group G.

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ΓK is 2-arc-transitive by Lemma 4.1, A¯ < Aut(ΓK ) does not necessarily act 2-arc-transitively on ΓK , and hence one cannot imply the 2-arc-transitivity of Γ by Theorem 2.4. The case where A¯ is quasiprimitive on V ΓK is discussed in the next lemma. Lemma 5.2. Assume A¯ is quasiprimitive on V ΓK . Then Γ is 2-arc-transitive. Proof. Since K is a cyclic group, subgroups of K are normal in A. Thus, to prove the lemma, by Theorem 2.4(2), it is sufficient to prove it when |K | is a prime. Assume K ∼ = Zm with m a prime. Suppose that, on the contrary, Γ is not 2-arc-transitive. Then ΓK is not (A¯ , 2)-arc-transitive by Theorem 2.4(2). By Lemma 4.1, ΓK is a complete graph, and A¯ is 2-primitive but not 3-transitive on V ΓK . Checking the candidates in Table 2 of Theorem 3.3, we conclude that A¯ = A4 , Z42 : A6 , Z42 : S6 , or ¯ = Z22 ▹ A, ¯ which contradicts that G¯ PSL(2, pe ) · o where pe ≡ 3 (mod 4) and o ≤ Ze . If A¯ = A4 , then G ¯ is core-free in A. Consider the remaining cases in the following. Let Y = K · soc(A¯ ). Then Y ▹ A. (a) Suppose soc(A¯ ) = Z42 . Then ΓK = K16 , A¯ = Z42 : A6 or Z42 : S6 , and Y = K · Z42 . Let X = K · Z42 · A6 ≤ A. Noting that ΓK is a (Z42 : A6 )-normal locally-primitive Cayley graph of Z42 , Γ is an X -normal locally-primitive Cayley graph of Y . Assume first that m is an odd prime. By Lemma 2.2, Y is nonabelian. Then, as Aut(K ) ∼ = Zm−1 is cyclic, we have that the centralizer CY (K ) = K · Z32 ∼ = Zm × Z32 and Y ∼ = (Zm × Z32 ) · Z2 . It follows that soc(Y ) ∼ = Zm × Z32 has a characteristic subgroup P ∼ = Z32 . Hence P is normal in X and Y /P ∼ = Zm · Z2 ∼ = D2m . Now, as P has 2m orbits on V Γ , by Theorem 2.4, the normal quotient graph ΓP is an X /P-normal locally-primitive Cayley graph of Y /P. Further, as X /P = (Y /P ) · A6 , the vertex stabilizer of X /P on V ΓP is isomorphic to A6 , then Lemma 2.2 implies that A6 ≤ Aut(Y /P ) ∼ = Aut(D2m ) ∼ = Zm : Zm−1 , which is a contradiction. Assume next m = 2. Then Y = Z2 · Z42 . If Y is abelian, by Lemma 2.2, Y ∼ = Z52 and A ≤ Z52 · S6 . 5 Since S6 has no element of order 8, Z2 · S6 has no element of order 16, which contradicts that D32 ∼ = G ≤ A ≤ Z52 · S6 . Thus Y is nonabelian. Let Z (Y ) denote the center of Y . Then Z (Y ) ̸= 1 and |Z (Y )| divides 8. If |Z (Y )| = 4 or 8, then Z (Y ) ▹ A has at least four orbits on V Γ , by Theorem 2.4, val(ΓZ (Y ) ) = val(Γ ) = 15, which is impossible as |V ΓZ (Y ) | ≤ 8. So |Z (Y )| = 2. Now, by Lemma 2.8, − + − 4 Y ∼ = D8 ∗ D8 or D8 ∗ Q8 , and Aut(Y ) ∼ = Z42 · O+ 4 (2) or Z2 · O4 (2) respectively, where O4 (2) and O4 (2) denote the orthogonal groups; however, as Γ is an X -normal locally-primitive Cayley graph of Y and X = Y · A6 , by Lemma 2.2, we have A6 ≤ Aut(Y ), yielding a contradiction. ¯ ∼ (b) Suppose soc(A¯ ) = PSL(2, pe ). By Theorem 3.3, G = Dpe +1 , Y = K · PSL(2, pe ), and A = Y · o, e ¯ we have Y ⊇ G. where p ≡ 3 (mod 4) and o ≤ Ze . Noting that soc(A¯ ) ⊇ G, Since K is cyclic, Y = K · PSL(2, pe ) is a central extension. By Lemma 2.5, Y = KY ′ . As p is odd, the Schur Multiplier of PSL(2, pe ) is Z2 , we have Y ′ = PSL(2, pe ) or SL(2, pe ). If m = 2, then G ∼ = D2(pe +1) , and Y = Z2 × PSL(2, pe ) or SL(2, pe ). For the former, as pe ≡ 3 (mod 4), Y has no element of order pe + 1; for the latter, Y has no dihedral subgroup by Lemma 2.9, both are not possible as Y ⊇ G. If m is an odd prime, then Y ′ = PSL(2, pe ) and Y = KY ′ = K × PSL(2, pe ). In particular, K is in the center of Y , and hence K is in the center of G as K ⊆ G ⊆ Y , which is not possible because the center of a dihedral group is Z2 .  The last lemma treats the case where A¯ is bi-quasiprimitive on V ΓK . Lemma 5.3. Assume A¯ is bi-quasiprimitive on V ΓK . Then Γ is 2-arc-transitive. Proof. As explained in Lemma 5.2, without loss of generality, we may suppose K ∼ = Zm with m a prime. ¯ Suppose that, by contradiction, Γ is not 2-arc-transitive. Then ΓK is an A-locally-primitive core¯ ¯ free Cayley graph of the dihedral group G but not (A, 2)-arc-transitive. By Lemma 4.1, A¯ and ΓK satisfy parts (iii)–(iv) of Lemma 3.7. Let α ∈ V Γ and let Y = K · soc(A¯ ). Suppose ΓK = Kq+1,q+1 − (q + 1)K2 , as in part (iii) of Lemma 3.7. Then A¯ = PGL(2, q) · Zf and ¯G = D2(q+1) , where q = pe ≡ 3 (mod 4) and f | e. Since G ∼ = D2m(q+1) is not contained in Y , and

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Y ∼ = K · PSL(2, q) has an overgroup K · PGL(2, q) containing G, we have G/(G ∩ Y ) ∼ = GY /Y ∼ = ∼ ∼ K · PGL(2, p)/Y = Z2 . Hence G ∩ Y = Zm(q+1) or Dm(q+1) . Assume m = 2. Then G ∩ Y ∼ = Z2(q+1) or D2(q+1) . Since the Schur Multiplier of PSL(2, q) is Z2 , Y = Z2 · PSL(2, q) ∼ = Z2 × PSL(2, q) or SL(2, q). However, since q ≡ 3 (mod 4), Z2 × PSL(2, q) has no subgroup isomorphic to Z2(q+1) or D2(q+1) , and by Lemma 2.9, so is SL(2, q), a contradiction occurs. Assume now m is an odd prime. Then Y = KY ′ ∼ = K × PSL(2, q) ∼ = Zm × PSL(2, q), and ∼ G ∩ Y = Zm(q+1) or Dm(q+1) . Obviously, Zm × PSL(2, q) has no subgroup isomorphic to Zm(q+1) . If G∩Y ∼ = Dm(q+1) , then K is in the center of the dihedral group G ∩ Y , which is also impossible. Finally, consider the case where ΓK = Kp,p , as in part (iv) of Theorem 3.7. Then G ∼ = D2mp and Y = K · Z2p . Since ⟨a⟩/K ⊆ Y /K , we have a ∈ Y , so Y = ⟨a, y⟩ = ⟨a⟩ : ⟨y⟩ for some element y of order p. Noting that Y ▹ A has exactly two orbits on V Γ and Yα ∼ = Zp is primitive on Γ (α), Γ is Y -vertexintransitive and edge-transitive. By Lemma 2.3, Γ ∼ = Cos(Y , Yα , Yβ ) for some β ∈ Γ (α). If Y is abelian, as Yα ∼ = Zp , ⟨Yα , Yβ ⟩ ̸= Y , Γ is not connected, which is a contradiction. Thus Y is nonabelian. = Yβ ∼ Let X = ⟨Y , b⟩. Then X acts vertex-transitively and locally-primitively on Γ . Let C = CX (K ) be the centralizer of K in X . Suppose ay = at . Then t p ≡ 1(mod km), and t ̸≡ 1(mod km) as Y is nonabelian. Suppose x = ai yj bs ∈ C . Then a = ax = a(−1) t , so (−1)s t j ≡ 1 (mod km), implying t 2j ≡ 1 (mod km). It follows that p | j and 2 | s, that is, x ∈ ⟨a⟩, Thus C = ⟨a⟩. Now, as X /⟨a⟩ ≤ Aut(⟨a⟩) is abelian, we have G/⟨a⟩ ▹ X /⟨a⟩, implying G ▹ X , that is, Γ is an X -normal locally-primitive Cayley graph of the dihedral group G. By Lemma 3.1, Γ ∼ = N D 2pm,p,k , where p | (m − 1). Since p ≥ 3, n = 2pm ≥ 42, again by Lemma 3.1, we conclude that Γ is a normal Cayley graph of G, which is a contradiction.  s j

Finally, we present a complete proof of Theorem 1.5. Proof of Theorem 1.5. If Γ is 2-arc-transitive, part (1) of Theorem 1.5 holds by Du et al. [7, Theorem 1.2]. Suppose that Γ is not 2-arc-transitive in the following. Let A = AutΓ and K = coreA (G). By Lemma 4.1, K ̸= 1. If K = G, by Lemma 3.1, Γ ∼ = N D 2n,r ,k , part (2) of Theorem 1.5 follows. If |G : K | = 2, then Γ = K4,4 or K4,4 − 4K2 by Lemma 5.1, both graphs are 2-arc-transitive, yielding a contradiction. ¯ = G/K . Then G¯ is core-free Assume that K ̸= 1 has at least three orbits on V Γ . Let A¯ = A/K and G ¯ and as |G : K | ≥ 3, G¯ is a regular dihedral group on V ΓN . By Theorem 2.4(2), ΓK is an A-locally¯ in A, ¯ and is not (A¯ , 2)-arc-transitive. It then follows from Lemma 4.1 primitive core-free dihedrant of G that A¯ is either quasiprimitive or bi-quasiprimitive on V ΓK . Now, Lemmas 5.2 and 5.3 imply that the graph Γ is 2-arc-transitive in both cases, which is a contradiction. This completes the proof of the theorem.  Acknowledgments The author is very grateful to the referees for their valuable comments. References [1] B. Alspach, M. Conder, D. Marušič, M.Y. Xu, A classification of 2-arc-transitive circulants, J. Algebraic Combin. 5 (1996) 83–86. [2] N. Biggs, Algebraic Graph Theory, second ed., Cambridge University Press, New York, 1992. [3] P.J. Cameron, Finite permutation groups and finite simple groups, Bull. London Math. Soc. 13 (1981) 1–22. [4] Y. Cheng, J. Oxley, On weakly symmetric graphs of order twice a prime, J. Combin. Theory Ser. B 42 (1987) 196–211. [5] M. Conder, C.H. Li, C.E. Praeger, On the Weiss conjecture for finite locally primitive graphs, Proc. Edinburgh Math. Soc. 43 (2000) 129–138. [6] J.H. Conway, R.T. Curtis, S.P. Noton, R.A. Parker, R.A. Wilson, Atlas of Finite Groups, Clarendon Press, Oxford, 1985. [7] S.F. Du, D. Malnič, D. Marušič, A classification of 2-arc-transitive dihedrants, J. Combin. Theory Ser. B 98 (2008) 1349–1372. [8] S.F. Du, D. Marušič, A.O. Waller, On 2-arc-transitive covers of complete graphs, J. Combin. Theory Ser. B 74 (1998) 276–290. [9] W.W. Fan, D. Leemans, C.H. Li, J.M. Pan, Locally 2-arc-transitive complete bipartite graphs, J. Combin. Theory Ser. A 120 (2013) 683–699. [10] W.W. Fan, C.H. Li, J.M. Pan, Finite locally primitive complete bipartite graphs, J. Group Theory (in press). [11] X.G. Fang, X.S. Ma, J. Wang, On locally primitive Cayley graphs of finite simple groups, J. Combin. Theory Ser. A 118 (2011) 1039–1051. [12] Y.Q. Feng, Y.T. Li, One-regular graphs of square-free order of prime valency, European J. Combin. 32 (2011) 261–275. [13] M. Giudici, C.H. Li, C.E. Praeger, Analysing finite locally s-arc-transitive graphs, Trans. Amer. Math. Soc. 356 (2003) 291–317. [14] C.D. Godsil, On the full automorphism group of a graph, Combinatorica 1 (1981) 243–256.

52 [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]

J. Pan / European Journal of Combinatorics 36 (2014) 39–52 I.M. Isaacs, Finite Group Theory, American Mathematics Society, 2008. A.A. Ivanov, C.E. Praeger, On finite affine 2-arc transitive graphs, European J. Combin. 14 (1993) 421–444. G. Jones, Cyclic regular subgroups of primitive permutation groups, J. Group Theory 5 (4) (2002) 403–407. W. Kantor, Classification of 2-transitive symmetric designs, Graphs Combin. 1 (1985) 165–166. C.H. Li, Finite edge-transitive Cayley graphs and rotary Cayley maps, Trans. Amer. Math. Soc. 358 (2006) 4605–4635. C.H. Li, J.M. Pan, Finite 2-arc-transitive abelian Cayley graphs, European J. Combin. 29 (2007) 148–158. C.H. Li, J.M. Pan, L. Ma, Locally primitive graphs of prime-power order, J. Aust. Math. Soc. 86 (2009) 111–122. C.H. Li, S.J. Song, D.J. Wang, Finite permutation groups with a regular dihedral subgroup, and edge-transitive dihedrants, submitted for publication. P. Lorimer, Vertex-transitive graphs: symmetric graphs of prime valency, J. Graph Theory 8 (1984) 55–68. D. Marušič, On 2-arc-transitivity of Cayley graphs, J. Combin. Theory Ser. B 87 (2003) 162–196. D. Marušič, Corrigendum to ‘‘on 2-arc-transitivity of Cayley graphs’’, J. Combin. Theory Ser. B 87 (2003) 162–196; J. Combin. Theory Ser. B 96 (2006) 761–764. J.M. Pan, Locally primitive normal Cayley graphs of metacyclic groups, Electron. J. Combin. 16 (2009) #R96. J.M. Pan, X. Yu, H. Zhang, Z.H. Huang, Finite edge-transitive dihedrant graphs, Discrete Math. 312 (2012) 1006–1012. C.E. Praeger, An O’Nan-Scott theorem for finite quasiprimitive permutation groups and an application to 2-arc transitive graphs, J. Lond. Math. Soc. 47 (1992) 227–239. D.J.S. Robinson, A Course in the Theory of Groups, Springer-Verlag, New York, 1982. D. Winter, The automorphism group of an extraspecial p-group, Rocky Mountain J. Math. 2 (2) (1972) 159–168.