Pentavalent symmetric graphs of order 2pqr

Discrete Mathematics 339 (2016) 522–532

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Pentavalent symmetric graphs of order 2pqr Da-Wei Yang, Yan-Quan Feng ∗ , Jia-Li Du Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China

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Article history: Received 20 November 2014 Received in revised form 28 July 2015 Accepted 11 September 2015

Keywords: Symmetric graph Arc-transitive One-regular Coset graph

abstract A graph is symmetric (or arc-transitive) if its automorphism group is transitive on the arc set of the graph. Let r < q < p be three distinct primes. In this paper, we give a complete classification of connected pentavalent symmetric graphs of order 2pqr with r ≥ 3. It is shown that a connected pentavalent symmetric graph Γ of order 2pqr is isomorphic to one of 20 sporadic coset graphs associated with some simple groups, a coset graph on the group PSL(2, p) or PGL(2, p) with p ≥ 29 and Aut(Γ ) ∼ = PSL(2, p) or PGL(2, p) respectively, or a 1-regular graph given in Feng and Li (2011). The connected pentavalent symmetric graphs of order 4pq have been classified before. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Let G be a permutation group on a set Ω and let α ∈ Ω . Denote by Gα the stabilizer of α in G, that is, the subgroup of G fixing the point α . We say that G is semiregular on Ω if Gα = 1 for every α ∈ Ω , and regular if G is transitive and semiregular. Denote by Zn , Dn , An and Sn the cyclic group of order n, the dihedral group of order 2n, the alternating group and the symmetric group of degree n, respectively. For a subgroup H of a group G, denote by CG (H ) the centralizer of H in G and by NG (H ) the normalizer of H in G. Throughout this paper, all groups and graphs are finite, and all graphs are simple and undirected. For a graph Γ , we denote its vertex set, edge set and automorphism group by V (Γ ), E (Γ ) and Aut(Γ ), respectively. For v ∈ V (Γ ), NΓ (v) is the set of neighbors of v in Γ , that is, the set of vertices adjacent to v in Γ . An s-arc in a graph Γ is an ordered (s + 1)-tuple (v0 , v1 , . . . , vs ) of vertices of Γ such that vi−1 is adjacent to vi for 1 ≤ i ≤ s, and vi−1 ̸= vi+1 for 1 ≤ i < s. A 1-arc is just called an arc. A graph Γ is said to be (G, s)-arc-transitive or (G, s)-regular if G acts transitively or regularly on the set of s-arcs of Γ , respectively, and (G, s)-transitive if G acts transitively on the s-arcs but not on the (s + 1)-arcs of Γ , where G ≤ Aut(Γ ) is a subgroup of Aut(Γ ). A graph Γ is said to be s-arc-transitive, s-regular or s-transitive if it is (Aut(Γ ), s)-arctransitive, (Aut(Γ ), s)-regular or (Aut(Γ ), s)-transitive, respectively. In particular, 0-arc-transitive means vertex-transitive, and 1-arc-transitive means arc-transitive or symmetric. A graph Γ is edge-transitive if Aut(Γ ) is transitive on the edge set E (Γ ). It is well known that a graph Γ is G-arc-transitive if and only if G is vertex-transitive and the vertex stabilizer Gv of v ∈ V (Γ ) in G is transitive on NΓ (v). Hence, the structure of the vertex stabilizer Gv plays an important role in the study of (G, s)-transitive graphs. For example, benefited from Djoković and Miller [7]’s result about the vertex stabilizers of cubic symmetric graphs, lots of works about classifications of cubic symmetric graphs were obtained by many authors (see [9,10, 24,25]). Due to the vertex stabilizers given in [27], symmetric tetravalent graphs have also been studied extensively in the literature (see [20,33]). For more results about vertex stabilizers of tetravalent symmetric graphs, one may refer [6,28,30]. Naturally, the next step is to characterize symmetric pentavalent graphs. Let Γ be a connected (G, s)-transitive pentavalent

∗

Corresponding author. E-mail addresses: [email protected] (D.-W. Yang), [email protected] (Y.-Q. Feng), [email protected] (J.-L. Du).

http://dx.doi.org/10.1016/j.disc.2015.09.009 0012-365X/© 2015 Elsevier B.V. All rights reserved.

D.-W. Yang et al. / Discrete Mathematics 339 (2016) 522–532

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Table 1 Automorphism groups of some pentavalent symmetric graphs of square-free order. Aut(Γ )

|V (Γ )|

Aut(Γ )v

Aut(Γ )

|V (Γ )|

Aut(Γ )v

S6 (S5 × S5 ) o Z2 PGL(2, 11) Aut(PSL(3, 4)) PGL(2, 11) PGL(2, 19)

2·3 2·5 2 · 11 2·3·7 2 · 3 · 11 2 · 3 · 19

S5 S4 × S5 A5 A0 L(2, 4) D10 A5

PSL(2, 41) PGL(2, 29) PGL(2, 59) PGL(2, 61) Aut(PSp(4, 4))

2 · 7 · 41 2 · 7 · 29 2 · 29 · 59 2 · 31 · 61 2 · 5 · 17

A5 A5 A5 A5 Z62 o 0 L(2, 4)

graph for some group G ≤ Aut(Γ ) and a positive integer s. The vertex stabilizer Gv of v ∈ V (Γ ) was determined by Weiss [30] for s ≥ 4 and by Guo and Feng [14] for s ≤ 3. Based on these results, Morgan [23] obtained edge stabilizers and arc stabilizers of pentavalent symmetric graphs. This encourages us to consider some work on pentavalent symmetric graphs. Let p, q, r be three distinct primes. Hua et al. determined the connected pentavalent symmetric graphs of order 2pq in [17]. Recently, Pan et al. [26] gave a classification of connected pentavalent symmetric graphs of order 4pq with p > q ≥ 5, and together with the works of Guo et al. [15] and Hua et al. [16], the connected pentavalent symmetric graphs of order 4pq with p > q ≥ 2 are known. In this paper, we classify connected pentavalent symmetric graphs of order 2pqr with p > q > r ≥ 3. Theorem 1.1. Let p > q > r be three distinct odd primes, and let Γ be a connected pentavalent symmetric graph of order 2pqr. Then one of the following holds. (1) Γ is one-regular and Γ ∼ = CD ℓpqr with 5(q − 1) and 5(p − 1) (defined in Section 2), and for a given order, either r = 5



  and there are exactly 4 non-isomorphic such graphs, or 5(r − 1) and there are 16 non-isomorphic such graphs. (2) Γ is 2-transitive and isomorphic to one of the following graphs, each of which is uniquely determined by its order. (2.1) Aut(Γ ) ∼ = PSL(2, p) with p ≡ 9, 39, 41, 71 (mod 80) and Γ ∼ = GPSL(2,p) (see Example 3.9); (2.2) Aut(Γ ) ∼ = PGL(2, p) with p ≡ 11, 19, 21, 29 (mod 40) and Γ ∼ = GPGL(2,p) (see Example 3.11); (j) (j) (i) (3) Γ is isomorphic to one of the following 20 sporadic graphs defined in Section 3: G390 , G1722 , G1218 , G2926 , G10,266 , G11,346 , where 1 ≤ i ≤ 3 and 1 ≤ j ≤ 7. The paper is organized as follows. After this introductory section, in Section 2 we will give some preliminary results. In Section 3, some connected pentavalent symmetric graphs are constructed, and in Section 4, the proof of Theorem 1.1 is given. 2. Preliminaries In this section, we describe some preliminary results which will be used later. First we describe vertex stabilizers of connected pentavalent symmetric graphs. By [14, Theorem 1.1] or [23, Theorem 1.2], we have the following proposition. Proposition 2.1. Let Γ be a connected pentavalent (G, s)-transitive graph for some G ≤ Aut(Γ ) and s ≥ 1. Let v ∈ V (Γ ). Then s ≤ 5, |Gv |29 · 32 · 5 and one of the following holds: (1) (2) (3) (4) (5)

For For For For For

s s s s s

= 1, Gv = 2, Gv = 3, Gv = 4, Gv = 5, Gv

∼ = Z5 , D5 or D10 ; ∼ = F20 , F20 × Z2 , A5 or S5 , where F20 is the Frobenius group of order 20; ∼ = F20 × Z4 , A4 × A5 , S4 × S5 or (A4 × A5 ) o Z2 with A4 o Z2 = S4 and A5 o Z2 = S5 ; ∼ = ASL(2, 4), AGL(2, 4), A6 L(2, 4) or A0 L(2, 4); ∼ = Z62 o 0 L(2, 4).

Let p and q be two distinct odd prime integers. Connected symmetric graphs of order 2p were classified by Cheng and Oxley [3] and Hua et al. [17] determined connected pentavalent symmetric graphs of order 2pq. In view of [3, Theorem 2.4] and [17, Theorem 4.2], one can get the following proposition. Proposition 2.2. Let Γ be a connected pentavalent symmetric graph of square-free order having at most three distinct prime divisors. Then either Aut(Γ ) is solvable, or Aut(Γ ) is isomorphic to one group listed in Table 1. Let Γ be a graph and N ≤ Aut(Γ ). The quotient graph ΓN of Γ relative to the orbits of N is defined as the graph with vertices the orbits of N on V (Γ ) and with two orbits adjacent if there is an edge in Γ between those two orbits. The theory of quotient graph is widely used to investigate symmetric graphs. Proposition 2.3 ([22, Theorem 9]). Let Γ be a connected G-arc-transitive graph of prime valency, and let N E G, a normal subgroup of G. If N has at least three orbits, then it is semiregular on V (Γ ) and ΓN is G/N-arc-transitive with G/N ≤ Aut(ΓN ).

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Let G be a finite group and H ≤ G. Denote by D a union of some double cosets of H in G such that D−1 = D. A coset graph Γ = Cos(G, H , D) on G with respect to H and D is defined to have vertex set V (Γ ) = [G : H ], the set of right cosets of H in G, and the edge set E (Γ ) = {{Hg , Hdg } | g ∈ G, d ∈ D}. Clearly, Cos(G, H , D) ∼ = Cos(G, H α , Dα ) for each α ∈ Aut(G). By [8], one can obtain the following proposition. Proposition 2.4. Let Γ be a connected G-arc-transitive graph, and let u ∈ NΓ (v). Then Γ is isomorphic to the coset graph g Cos(G, Gv , Gv gGv ), where g is a 2-element in G such that Guv = Guv , g 2 ∈ Gv and ⟨Gv , g ⟩ = G. Moreover, Γ has valency g |Gv : Gv ∩ Gv |. Let S be a generator subset of G with 1 ̸∈ S and S −1 = S. Clearly, the coset graph Γ = Cos(G, 1, S ) is a connected undirected simple graph, which is called a Cayley graph on G with respect to S and denoted by Cay(G, S ). Let R(G) be the right regular representation of G, the acting group of G by right multiplication. Then R(G) ≤ Aut(Γ ). By Godsil [12] or Xu [32], NAut(Γ ) (R(G)) = R(G) o Aut(G, S ), where Aut(G, S ) = {α ∈ Aut(G) | S α = S }. A Cayley graph Γ = Cay(G, S ) is said to be normal if R(G) is normal in Aut(Γ ), and in this case, Aut(Γ ) = R(G) o Aut(G, S ). The following infinite family of Cayley graphs was first constructed in [19]. Let m and ℓ be two integers such that m ≥ 31 and ℓ4 + ℓ3 + ℓ2 + ℓ + 1 ≡ 0 (mod m). Let

CD ℓm = Cay(Dm , {b, ab, aℓ+1 b, aℓ

2 +ℓ+1

3 2 b, aℓ +ℓ +ℓ+1 b})

(1)

be a Cayley graph on the dihedral group Dm = ⟨a, b | a = b = 1, a = a ⟩. It is proved in [19, Theorem B and Proposition 4.1] that a pentavalent Cayley graph on a dihedral group Dm is one-regular if and only if it is isomorphic to one of CD ℓm for pairs (m, ℓ) such that m ≥ 31 and ℓ4 + ℓ3 + ℓ2 + ℓ + 1 ≡ 0 (mod m), and such a Cayley graph is normal, that is, Aut(CD ℓm ) ∼ = Dm o Z5 . From [11, Theorem 3.6], pentavalent one-regular graphs of square-free order were classified as follows: m

2

b

−1

Proposition 2.5. Let m be an odd square-free integer, and let Γ be a connected pentavalent one-regular graph of order 2m. Then m = 5t p1 p2 · · · ps ≥ 31 and Γ ∼ = CD ℓm for some ℓ such that ℓ4 + ℓ3 + ℓ2 + ℓ + 1 ≡ 0 (mod m), where t ≤ 1, s ≥ 1 and pi ’s  are distinct primes such that 5 (pi − 1). Furthermore, there are exactly 4s−1 non-isomorphic such graphs for a given order. In the end of this section, we introduce some results associated with finite groups. In view of [18, Theorem 4.2], we have the following. Proposition 2.6. Let F be the Fitting subgroup of a group G. If G is solvable, then CG (F ) ≤ F . The following proposition comes from [5, Section 239] and [2, Theorem 2] (also see [17, Propositions 2.4 and 2.5]). Proposition 2.7. Let p ≥ 5 be a prime. Then a maximal subgroup of PSL(2, p) is isomorphic to one of the following groups: (1) D p−1 with p ̸= 5, 7, 9, 11; 2

(2) D p+1 with p ̸= 7, 9; 2

(3) Zp o Z p−1 ; 2

(4) A4 with p = 5 or p ≡ 3, 13, 27, 37 (mod 40); (5) S4 with p ≡ ±1 (mod 8); (6) A5 with p ≡ ±1 (mod 5). A maximal subgroup of PGL(2, p) is isomorphic to one of the following groups: (1) (2) (3) (4) (5)

Dp−1 with p ≥ 7; Dp+1 ; Zp o Zp−1 ; S4 with p ≡ ±3 (mod 8); PSL(2, p).

Let G and E be two groups. We call an extension E of G by N a central extension of G if E has a central subgroup N such that E /N ∼ = G, and if further E is perfect, that is, if it equals its derived subgroup E ′ , we call E a covering group of G. Schur [29] proved that for every non-abelian simple group G there is a unique maximal covering group M such that every covering group of G is a factor group of M (also see [18, V §23]). This group M is called the full covering group of G, and the center of M is the Schur multiplier of G, denoted by Mult(G). Lemma 2.8. Let G be a group, and let N be an abelian normal subgroup of G such that Aut(N ) is solvable. If G/N is a non-abelian simple group, then G = G′ N and G′ ∩ N . Mult(G/N ).

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Table 2 Non-abelian simple groups with restrictive orders. G (∼ =)

Order

Mult(G)

A5 A6 A7 A8 PSL(2, 11) PSL(2, 24 ) PSL(2, 19) PSL(3, 4) PSL(2, 25) PSp(4, 4) PSU(3, 4) Sz (8)

22 23 23 26 22 24 22 26 23 28 26 26

Z2 Z6 Z6 Z2 Z2 1 Z2 Z4 × Z12 Z2 1 1 Z2 × Z2

·3·5 · 32 · 5 · 32 · 5 · 7 · 32 · 5 · 7 · 3 · 5 · 11 · 3 · 5 · 17 · 32 · 5 · 19 · 32 · 5 · 7 · 3 · 52 · 13 · 32 · 52 · 17 · 3 · 52 · 13 · 5 · 7 · 13

G(∼ =)

Order

Mult(G)

Sz (32) M11 M12 M22 M23 J1 J2 PSL(5, 2) PSL(2, 26 ) PSL(2, 28 ) PSL(2, t )

210 · 52 · 31 · 41 24 · 32 · 5 · 11 26 · 33 · 5 · 11 27 · 32 · 5 · 7 · 11 27 · 32 · 5 · 7 · 11 · 23 23 · 3 · 5 · 7 · 11 · 19 27 · 33 · 52 · 7 210 · 32 · 5 · 7 · 31 26 · 32 · 5 · 7 · 13 28 · 3 · 5 · 17 · 257 t ≥ 29 and t ∈ {r , q, p}

1 1 Z2 Z12 1 1 Z2 1 1 1 Z2

Proof. Let C = CG (N ). As N is abelian, N ≤ C . Since G/C . Aut(N ) is solvable and G/N is a non-abelian simple group, C /N is insolvable. It follows that C /N = G/N. Thus, G = C is a central extension of G/N with center N. Since G/N = G′ N /N ∼ = G′ /(G′ ∩ N ), we have G = G′ N and since G′ = (G′ N )′ = (G′ )′ , G′ is a covering group of G/N. Hence, ′ G ∩ N . Mult(G/N ).  The following lemma is proved by the well-known finite simple group classification (see [13, 2.12] or [31, 1.2]). Lemma 2.9. Let r < q < p be three distinct odd prime integers, and let G be a non-abelian simple group such that 5|G| and



 |G|210 · 32 · 5 · r · q · p. Then G is isomorphic to one of the groups listed in Table 2. Proof. Clearly, we have 211 - |G|,

34 - |G|,

53 - |G|,



5|G|,

and t 2 - |G|

(2)

  where t ∈ {r , q, p} and t ≥ 7. Specially, if 52 |G| or 33 |G|, then |G| has at most five prime divisors. By [31, 1.2, pp. 3], each finite non-abelian simple group is isomorphic to An with n ≥ 5, one of 26 sporadic simple groups, or a classical group or an exceptional group of Lie type. For the orders of these simple groups, one can see [13, Table 2.4, pp. 134–136], and for more details, see [31, Sections 3, 4, 5]. For An with n ≥ 5, since 34 - |G|, we have G ∼ = A5 , A6 , A7 or A8 . For the 26 sporadic simple groups, by Eq. (2) we have ∼ G = M11 , M12 , M22 , M23 , M24 , J1 or J2 . Since 33 |M24 | and |M24 | has 6 prime divisors, we have G  M24 . For the groups of Lie type, since 211 - |G| and each odd prime divisor of |G| has power at most 3, by [13, Table 2.4, pp. 134–136], G ∼ = PSL(n, t ) with n ≥ 2, PSU(n, t ) with n ≥ 3, PSp(2n, t ) with n ≥ 2, or Sz (22n+1 ) with n ≥ 1, where t is a prime power. Let G ∼ = PSL(n, t ). Then |G| = (n,t1−1) t n(n−1)/2 Πin=2 (t i − 1). First assume n ≥ 3. Then n(n − 1)/2 ≥ 3, and by Eq. (2), we have n = 3 and t = 3 or 2i with i ≤ 3, n = 4 and t = 2, or n = 5 and t = 2. For each case, by checking orders with Eq. (2) again, we have G ∼ = PSL(3, 4), PSL(4, 2)(∼ = A8 ) or PSL(5, 2). Now assume n = 2. Then |G| = (2,t1−1) t (t 2 − 1). If

t = 2i , then i ≤ 10 by Eq. (2). Similarly, if t = 3i , then i ≤ 3; if t = 5i , then i ≤ 2; if t = si with s ≥ 7 and s ∈ {r , q, p}, then i = 1. For each case, checking the orders of PSL(2, t ) again, we have G ∼ = PSL(2, 22 ) ∼ = A5 , PSL(2, 24 ), PSL(2, 26 ), PSL(2, 28 ), PSL(2, 9)(∼ = A6 ), PSL(2, 5)(∼ = A5 ), PSL(2, 25), PSL(2, 11), PSL(2, 19), or PSL(2, t ) with some prime t ≥ 29 and t ∈ {r , q, p}. Let Γ ∼ = PSU(n, t ) with n ≥ 3. Then |G| = (n,t1+1) t n(n−1)/2 Πin=2 (t i − (−1)i ). Since n(n − 1)/2 ≥ 3, we have n = 3 and

t = 3 or t = 2i with i ≤ 3, or n = 4 or 5 and t = 2 by Eq. (2). Hence, G ∼ = PSU(3, 4). For other two infinite families 2

PSp(2n, t ) of order (n,t1−1) t n Πin=1 (t 2i − 1) with n ≥ 2 and Sz (22n+1 ) of order 24n+2 (24n+2 + 1)(22n+1 − 1) with n ≥ 1, one may similarly obtain that G ∼ = PSp(4, 4), Sz (8) or Sz (32). The Schur multipliers of these simple groups can be obtained from [13, Table 4.1].  3. Constructions as Coset graphs In this section, we introduce some pentavalent symmetric graphs which are constructed as coset graphs. The following graph was first constructed in [21, Construction 3.3]. Example 3.1. Let G = PSL(2, 25). By [21, Construction 3.3], G has a subgroup H ∼ = F20 and an involution g such that |H : H ∩ H g | = 5 and ⟨H , g ⟩ = G. Denote by G390 the coset graph Cos(G, H , HgH ). By [21, Lemma 3.4], G390 is a connected pentavalent 2-regular graph with Aut(Γ ) ∼ = PSL(2, 25), and each connected pentavalent symmetric graph Γ of order 390 admitting PSL(2, 25) as an arc-transitive automorphism group is isomorphic to G390 . Example 3.2. Let G = J1 . By Atlas [4, pp. 36], G has a subgroup H ∼ = A5 and an involution g ∈ G such that |HgH |/|H | = 5 and ⟨H , g ⟩ = G. Denote by G2926 the coset graph Cos(G, H , HgH ).

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Lemma 3.3. Each connected pentavalent symmetric graph Γ of order 2926 admitting J1 as an arc-transitive automorphism group is isomorphic to G2926 . Furthermore, Γ is 2-transitive and Aut(Γ ) ∼ = J1 . Proof. Let G = J1 . As Γ is a G-arc-transitive graph of order 2926, |Gv | = 60 for any vertex v ∈ V (Γ ) and by Proposition 2.1, we have Gv ∼ = A5 . By Proposition 2.4, Γ ∼ = Cos(G, Gv , Gv fGv ) for some 2-element f such that f 2 ∈ Gv , ⟨Gv , f ⟩ = G and |Gv : Gv ∩ Gfv | = 5. Since all such subgroups Gv are conjugate in G by MAGMA [1], we may assume that Gv = H, and since A5 has only five subgroups isomorphic to A4 , we may further assume that H ∩ H f = H ∩ H g (if necessary, replace f by f h for some h ∈ H), where H and g are given in Example 3.2. Set L = H ∩ H g . Then f ∈ NG (L), and by MAGMA [1], we have NG (L) ∼ = Z2 × A4 . Since ⟨H , g ⟩ = ⟨H , f ⟩ = G, we have g , f ̸∈ L and thus NG (L) = L ∪ Lg = L ∪ Lf . Hence, Lg = Lf and f = ℓg for some ℓ ∈ L. It follows that HfH = H ℓgH = HgH and Γ ∼ = Cos(G, H , HgH ). Again by MAGMA [1], we have Aut(Γ ) ∼ = J1 (also see Lemmas 4.1 and 4.2).  Example 3.4. Let G ∼ = PSL(2, 29) be a subgroup of S30 such that G contains the following elements: a = (1 7)(2 12)(3 23)(4 28)(5 19)(6 17)(8 21)(10 11)(13 15)(14 27) (16 30)(18 29)(20 22)(24 25), b = (1 7 30 9 16)(2 26 12 15 13)(3 28 11 25 8)(4 23 21 24 10) (5 20 27 6 29)(14 22 19 18 17), d1 = (1 23)(2 13)(3 7)(4 29)(5 25)(8 11)(9 26)(10 21)(12 15)(14 27) (16 20)(18 28)(19 24)(22 30), d2 = (1 10)(2 19)(3 30)(4 24)(5 12)(6 8)(7 11)(9 26)(14 29)(16 23) (17 21)(18 27)(20 22)(25 28), d3 = (1 20)(3 8)(4 27)(5 18)(6 11)(7 22)(9 26)(10 17)(13 24)(14 28) (15 25)(16 30)(19 29)(21 23). By MAGMA [1], G = ⟨a, b, di ⟩ for each integer 1 ≤ i ≤ 3 and H := ⟨a, b⟩ ∼ = D5 . Define three coset graphs: (i)

G1218 = Cos(G, H , Hdi H ),

i = 1, 2, 3. (1)

(2)

(3)

Again by MAGMA [1], the three coset graphs G1218 , G1218 and G1218 are pairwise non-isomorphic connected pentavalent (1)

(2) (3) 1-transitive graphs of order 1218 with Aut(G1218 ) ∼ = Aut(G1218 ) ∼ = PSL(2, 29) and Aut(G1218 ) ∼ = PGL(2, 29).

Lemma 3.5. Each connected pentavalent symmetric graph of order 1218 admitting a group PSL(2, 29) as an arc-transitive (i) automorphism group is isomorphic to one of G1218 with 1 ≤ i ≤ 3. Proof. Let G ∼ = PSL(2, 29) and let Γ be a connected pentavalent symmetric graph of order 1218 admitting G as an arctransitive automorphism group. Then |Gv | = 10 for each vertex v ∈ V (Γ ), and by Proposition 2.1, we have Gv ∼ = D5 . By Proposition 2.4, Γ ∼ = Cos(G, Gv , Gv fGv ) for some 2-element f ∈ G such that f 2 ∈ Gv , ⟨f , Gv ⟩ = G and |Gv , Gv ∩ Gfv | = 5. Since all the subgroups isomorphic to D5 are conjugate in G, by Proposition 2.4, we may assume that Gv = H (see Example 3.4). By MAGMA [1], for each 2-element f in G such that f 2 ∈ H, ⟨f , H ⟩ = G and |H : H ∩ H f | = 5, the corresponding coset graph (i) Cos(G, H , HfH ) is isomorphic to one of G1218 , where 1 ≤ i ≤ 3.  Similarly as Example 3.4 and Lemma 3.5, by MAGMA [1], one can construct several symmetric pentavalent graphs as coset graphs on the groups PSL(2, 41), PSL(2, 59), PSL(2, 61), PGL(2, 59) and PGL(2, 61), as in the following three examples. Example 3.6. Let G ∼ = PSL(2, 41) be a subgroup of S42 such that G contains the following elements: a = (1 36 30 15 39 21 6 42 35 18)(2 14 29 7 22 34 3 5 31 33) (4 23 11 27 16 37 17 40 8 12)(9 25 13 32 24 28 38 19 41 20), b = (1 31)(2 35)(3 30)(4 40)(5 36)(6 29)(7 21)(8 12)(9 13)(10 26)(11 37) (14 42)(15 34)(16 27)(17 23)(18 33)(19 28)(20 32)(22 39)(24 41), d = (1 28)(2 9)(3 29)(4 36)(5 39)(6 12)(7 8)(10 13)(11 40)(14 35)(15 18) (16 22)(17 30)(19 26)(20 21)(23 31)(24 34)(25 41)(27 42)(33 37). By MAGMA [1], G = ⟨a, b, d⟩ and H := ⟨a, b⟩ ∼ = D10 . Denote by G1722 the coset graph Cos(G, H , HdH ). Again by MAGMA [1], the graph G1722 is a connected 1-transitive pentavalent graph of order 1722 with Aut(G1722 ) ∼ = PSL(2, 41) and each connected pentavalent symmetric graph of order 1722 admitting PSL(2, 41) as an arc-transitive automorphism group is isomorphic to G1722 .

ˆ be two subgroups of S60 such that G ∼ Example 3.7. Let G ≤ G = PSL(2, 59), Gˆ ∼ = PGL(2, 59) and Gˆ contains the following elements: a = (1 59 14 5 53 25 4 58 9 38)(2 31 42 36 15 47 35 26 41 39) (3 37 27 60 12 54 46 28 40 13)(6 49 16 44 20 56 24 51 34 50) (7 17 10 48 45 29 43 18 21 8)(11 55 52 30 23 33 32 19 22 57);

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b = (1 29)(2 52)(3 49)(4 8)(5 10)(6 37)(7 25)(9 18)(11 42)(12 51)(13 16) (14 48)(15 22)(17 53)(19 47)(20 28)(21 58)(23 41)(24 54)(26 33)(27 50) (30 39)(31 55)(32 35)(34 60)(36 57)(38 43)(40 44)(45 59)(46 56); d1 = (1 56)(2 42)(3 44)(4 48)(5 55)(6 15)(7 51)(8 14)(9 26)(10 31)(11 52) (12 25)(13 53)(16 17)(18 33)(19 23)(20 58)(21 28)(22 37)(24 45)(27 34) (29 46)(30 43)(32 36)(35 57)(38 39)(40 49)(41 47)(50 60)(54 59); d2 = (1 33)(2 37)(3 11)(4 22)(5 51)(6 52)(7 44)(8 15)(9 60)(10 12)(13 50) (14 36)(16 27)(17 32)(18 34)(19 59)(20 55)(21 43)(23 39)(24 56)(25 40) (26 29)(28 31)(30 41)(35 53)(38 58)(42 49)(45 47)(46 54)(48 57); d3 = (1 45)(2 40)(3 51)(4 35)(5 58)(6 14)(7 55)(8 32)(9 46)(10 21)(11 15) (12 49)(13 57)(16 36)(17 34)(18 56)(19 38)(20 39)(22 42)(23 54)(24 41) (25 31)(26 50)(27 33)(28 30)(29 59)(37 48)(43 47)(44 52)(53 60); d4 = (1 41)(2 12)(3 55)(4 13)(5 39)(6 32)(7 58)(8 16)(9 59)(10 30)(11 14) (15 44)(17 33)(18 45)(19 20)(21 25)(22 40)(23 29)(24 43)(26 53)(27 56) (28 47)(31 49)(34 36)(35 37)(38 54)(42 48)(46 50)(51 52)(57 60); d5 = (1 43)(2 31)(3 58)(4 60)(5 19)(6 16)(7 39)(8 34)(9 23)(10 47)(11 32) (12 22)(13 37)(14 44)(15 51)(17 56)(18 41)(20 54)(21 49)(24 28)(25 30) (26 57)(27 45)(29 38)(33 36)(35 42)(40 48)(46 53)(50 59)(52 55); d6 = (1 28)(2 21)(3 39)(4 26)(5 54)(6 34)(7 19)(8 33)(9 38)(10 24)(11 16) (12 48)(13 42)(14 51)(15 55)(17 45)(18 43)(20 29)(22 31)(23 50)(25 47) (27 41)(30 49)(32 44)(35 40)(36 56)(37 60)(46 57)(52 58)(53 59); d = (1 18)(2 38)(3 28)(4 58)(5 25)(6 31)(7 10)(8 21)(9 29)(11 12)(13 26) (14 45)(15 41)(16 33)(17 47)(19 53)(20 49)(22 23)(24 27)(30 36)(32 56) (35 46)(37 55)(39 57)(40 44)(42 51)(43 52)(48 59)(50 54).

ˆ = ⟨a, b, d⟩, H := ⟨a2 , b⟩ ∼ By MAGMA [1], we have G = ⟨a2 , b, di ⟩ for each integer 1 ≤ i ≤ 6, G = D10 . = D5 and Hˆ := ⟨a, b⟩ ∼ Define the following seven coset graphs: (i)

G10,266 = Cos(G, H , Hdi H ),

1 ≤ i ≤ 6,

(7)

ˆ Hˆ ). G10,266 = Cos(Gˆ , Hˆ , Hd (i)

(7)

Again by MAGMA [1], the graphs G10,266 and G10,266 are pairwise non-isomorphic connected pentavalent 1-transitive graphs (i)

(7)

of order 10,266(=2 · 3 · 29 · 59) with Aut(G10,266 ) ∼ = PSL(2, 59) and Aut(G10,266 ) ∼ = PGL(2, 59), where 1 ≤ i ≤ 6. Furthermore, each connected pentavalent symmetric graph of order 10,266 admitting PSL(2, 59) (or PGL(2, 59), resp.) as an arc-transitive (i) (7) automorphism group is isomorphic to one of G10,266 with 1 ≤ i ≤ 6 (or G10,266 , resp.).

ˆ be two subgroups of S62 such that G ∼ Example 3.8. Let G ≤ G = PSL(2, 61), Gˆ ∼ = PGL(2, 61) and Gˆ contains the following elements: a = (1 48 12 32 55 57 35 8 39 36)(2 27 56 30 14 24 7 46 3 60) (4 11 54 29 23 19 40 49 17 25)(5 26 22 16 53 34 41 20 28 58) (6 37 10 50 52 13 33 59 44 9)(15 51 18 43 47 42 61 38 21 31); b = (1 30)(2 32)(3 57)(4 54)(5 16)(6 51)(7 8)(9 18)(10 31)(12 27)(13 61) (14 36)(15 37)(17 23)(19 49)(20 41)(21 50)(22 26)(24 39)(25 29)(28 34) (33 42)(35 46)(38 52)(43 44)(45 62)(47 59)(48 56)(53 58)(55 60); d = (1 59)(2 28)(3 10)(4 46)(5 24)(6 14)(7 17)(8 23)(9 37)(11 40)(12 29) (13 55)(15 18)(16 39)(19 48)(20 43)(21 26)(22 50)(25 27)(30 47)(31 57) (32 34)(33 38)(35 54)(36 51)(41 44)(42 52)(45 53)(49 56)(58 62)(60 61); d1 = (2 17)(3 51)(4 26)(5 27)(6 57)(7 13)(8 61)(9 39)(10 31)(11 40)(12 16) (14 29)(15 19)(18 24)(20 53)(21 62)(22 54)(23 32)(25 36)(28 42)(33 34) (35 52)(37 49)(38 46)(41 58)(43 48)(44 56)(45 50)(47 55)(59 60); d2 = (1 42)(2 60)(3 49)(4 15)(5 51)(6 16)(7 8)(9 31)(10 18)(11 40)(12 53) (13 26)(14 43)(17 62)(19 57)(20 34)(21 24)(22 61)(23 45)(27 58)(28 41) (30 33)(32 55)(35 59)(36 44)(37 54)(38 48)(39 50)(46 47)(52 56); d3 = (1 28)(2 21)(3 27)(4 7)(5 41)(6 15)(8 54)(9 45)(10 39)(11 40)(12 57) (13 61)(16 20)(17 59)(18 62)(19 26)(22 49)(23 47)(24 31)(25 56)(29 48) (30 34)(32 50)(33 53)(35 44)(37 51)(38 55)(42 58)(43 46)(52 60); d4 = (1 41)(2 35)(3 54)(4 57)(5 49)(6 53)(7 22)(8 26)(9 18)(10 21)(11 40) (12 34)(13 15)(14 38)(16 19)(17 24)(20 30)(23 39)(25 44)(27 28)(29 43) (31 50)(32 46)(36 52)(37 61)(45 55)(47 48)(51 58)(56 59)(60 62); d5 = (1 58)(2 18)(3 41)(4 61)(5 42)(6 26)(7 19)(8 49)(9 32)(10 45)(11 40)

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(12 15)(13 54)(14 48)(16 33)(17 52)(20 57)(21 59)(22 51)(23 38)(24 39) (25 35)(27 37)(29 46)(30 53)(31 62)(36 56)(43 55)(44 60)(47 50); d6 = (1 37)(2 45)(3 28)(4 51)(5 58)(6 54)(7 20)(8 41)(9 43)(10 38)(11 40) (12 61)(13 27)(14 23)(15 30)(16 53)(17 36)(18 44)(19 49)(21 25)(22 42) (24 47)(26 33)(29 50)(31 52)(32 62)(34 57)(39 59)(48 55)(56 60). ˆ = ⟨a, b, d⟩, H := ⟨a2 , b⟩ ∼ By MAGMA [1], G = ⟨a2 , b, di ⟩ for each integer 1 ≤ i ≤ 6, G = D5 and Hˆ := ⟨a, b⟩ ∼ = D10 . Define the following seven coset graphs: (i)

G11,346 = Cos(G, H , Hdi H ),

1 ≤ i ≤ 6,

(7)

ˆ Hˆ ). G11,346 = Cos(Gˆ , Hˆ , Hd (i)

(7)

Again by MAGMA [1], the graphs G11,346 and G11,346 are pairwise non-isomorphic connected pentavalent 1-transitive graphs (i)

(7)

of order 11,346(=2 · 3 · 31 · 61) with Aut(G11,346 ) ∼ = PSL(2, 61) and Aut(G11,346 ) ∼ = PGL(2, 61), where 1 ≤ i ≤ 6. Furthermore, every connected pentavalent symmetric graph of order 11,346 admitting PSL(2, 61) (or PGL(2, 61), resp.) as an arc-transitive (i) (7) automorphism group is isomorphic to one of G11,346 with 1 ≤ i ≤ 6 (or G11,346 , resp.). In the end of this section, we construct two infinite families of arc-transitive graphs associated with the simple group PSL(2, p). In each family, there is a unique graph for a given order. Example 3.9. Let p be a prime such that p ≡ 1, 9, 31, 39 (mod 40) and let G = PSL(2, p). Then G has a subgroup H ∼ = A5 and an involution g such that ⟨H , g ⟩ = G and |H : H ∩ H g | = 5. Denote by GPSL(2,p) the coset graph Cos(G, H , HgH ). Lemma 3.10. Each connected pentavalent symmetric graph Γ admitting PSL(2, p) as an arc-transitive automorphism group with vertex stabilizer isomorphic to A5 is isomorphic to GPSL(2,p) , where p is a prime and p ≡ 1, 9, 31, 39 (mod 40). Proof. Let G = PSL(2, p) with p ≡ 1, 9, 31, 39 (mod 40). By [2, Theorem 2(vi)], all the subgroups isomorphic to A5 have two conjugate classes in G, and let H1 and H2 be two subgroups isomorphic to A5 in G that are not conjugate. Let Li ∼ = A4 be a subgroup of Hi , where i = 1 or 2. Then NHi (Li ) = Li and again by [2, Theorem 2(vi)], we have NG (Li ) ∼ = S4 , and thus NG (Li ) = Li ∪ Li gi for some involution gi ∈ NG (Li )\Li . Since Hi ∼ = A5 is a maximal subgroup of G, we have ⟨Hi , gi ⟩ = G and g Hi ∩ Hi i = Li . Hence, the coset graph Cos(G, Hi , Hi gi Hi ) is a connected pentavalent symmetric graph. f

First, we prove that for any 2-element fi in G such that fi2 ∈ Hi , ⟨Hi , fi ⟩ = G and |Hi : Hi ∩ Hi i | = 5, we have Cos(G, Hi , Hi fi Hi ) = Cos(G, Hi , Hi gi Hi ), where i = 1 or 2. Noting that there is only one conjugate class of subgroups f g isomorphic to A4 in Hi ∼ = A5 , we may assume that Hi ∩ Hi i = Hi ∩ Hi i = Li (if necessary, replace fi by fih for some h ∈ Hi ). Since fi2 ∈ Hi , we have fi ∈ NG (Li ) = Li ∪ Li gi and since fi ̸∈ Li , fi = ℓi gi for some ℓi ∈ Li . It follows that Hi fi Hi = Hi ℓi gi Hi = Hi gi Hi . Hence, Cos(G, Hi , Hi fi Hi ) = Cos(G, Hi , Hi gi Hi ). To prove Γ ∼ = GPSL(2,p) , by Proposition 2.4 and above argument, we only need to prove that Cos(G, H1 , H1 g1 H1 ) ∼ = Cos(G, H2 , H2 g2 H2 ). Note that Aut(G) ∼ = PGL(2, p) by [31, Theorem 3.2], and by [2, Theorem 2(vi)], all the subgroups isomorphic to A5 are conjugate in PGL(2, p) and they are contained in a subgroup PSL(2, p). Hence, there exists an automorphism α ∈ Aut(G) such that H2 = H1α . It follows that Cos(G, H1 , H1 g1 H1 ) ∼ = Cos(G, H1α , H1α g1α H1α ) = Cos(G, H2 , H2 g1α H2 ), and by the argument in above paragraph, we have Cos(G, H1 , H1 g1 H1 ) ∼ = Cos(G, H2 , H2 g2 H2 ), as required.  Example 3.11. Let p be a prime such that p ≡ 11, 19, 21, 29 (mod 40), and let G ∼ = PGL(2, p). Then G has a subgroup H ∼ = A5 and an involution g such that ⟨H , g ⟩ = G and |H : H ∩ H g | = 5. Denote by GPGL(2,p) the coset graph Cos(G, H , HgH ). Lemma 3.12. Each connected pentavalent symmetric graph Γ admitting PGL(2, p) as an arc-transitive automorphism group with vertex stabilizer isomorphic to A5 is isomorphic to GPGL(2,p) , where p is a prime and p ≡ 11, 19, 21, 29 (mod 40). Proof. Let G = PGL(2, p) with p ≡ 11, 19, 21, 29 (mod 40), and let v ∈ V (Γ ). Then Gv ∼ = A5 . Set L ∼ = A4 be a subgroup of Gv . By [2, Theorem 2(vi)], NG (L) ∼ = S4 and it is a maximal subgroup of G. This implies that there is an involution g ∈ NG (L)\L such that NG (L) = L ∪ Lg, and moreover ⟨Gv , g ⟩ = G and Gv ∩ Ggv = L. Hence, the coset graph Cos(G, Gv , Gv gGv ) is a connected pentavalent symmetric graph. Note that all subgroups isomorphic to A5 are conjugate in PGL(2, p) by [2, Theorem 2(vi)]. To prove Γ ∼ = GPGL(2,p) , by Proposition 2.4, we only need to prove that Cos(G, Gv , Gv fGv ) = Cos(G, Gv , Gv gGv ) for each 2-element f ∈ G such that f 2 ∈ Gv , ⟨Gv , f ⟩ = G and |Gv : Gv ∩ Gfv | = 5. Since A5 has exactly 5 subgroups isomorphic to A4 , we may assume that Gv ∩ Gfv = Gv ∩ Ggv = L. Hence, f ∈ NG (L) = L ∪ Lg and since f ̸∈ L, we have f = ℓg for some ℓ ∈ L. This implies that Gv fGv = Gv ℓgGv = Gv gGv and Cos(G, Gv , Gv fGv ) = Cos(G, Gv , Gv gGv ), as required.  4. Proof of Theorem 1.1 In this section, we always assume that Γ is a connected pentavalent symmetric graph of order 2pqr, where r < q < p are three distinct odd primes. Then r ≥ 3, q ≥ 5 and p ≥ 7. Let A = Aut(Γ ) and v ∈ V (Γ ). By Proposition 2.1, |Av | = 2i · 3j · 5

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and thus |A| = 2i+1 · 3j · 5 · r · q · p, where 0 ≤ i ≤ 9 and 0 ≤ j ≤ 2. To prove Theorem 1.1, we need the following two lemmas. Lemma 4.1. Assume that each minimal normal subgroup of A is solvable. Then Γ is 1-regular. Proof. For a prime divisor t of |A|, let Ot (A) be the largest normal t-subgroup of A. Since |V (Γ )| = 2pqr with 3 ≤ r < q < p, Ot (A) has  at least three orbits on V (Γ ). It follows from Proposition 2.3 that Ot (A) is semiregular on V (Γ ) and thus |Ot (A)|2pqr. Hence, |Ot (A)| = 1 or t. In particular, ΓOt (A) is a connected pentavalent A/Ot (A)-arc-transitive graph of order 2pqr /|Ot (A)|, yielding that |O2 (A)| = 1. Since each minimal normal subgroup of A is solvable, one may choose some t ∈ {p, q, r } such that Ot (A) ̸= 1, that is, Ot (A) = Zt . Suppose that A is insolvable. Then A/Ot (A) is insolvable and thus Aut(ΓOt (A) ) is insolvable because A/Ot (A) ≤ Aut(ΓOt (A) ). Since |V (ΓOt (A) )| = 2pqr /t is square-free and has exactly three distinct prime divisors, by Proposition 2.2, Aut(ΓOt (A) ) = Aut(PSL(3, 4)), PGL(2, 11), PGL(2, 19), PSL(2, 41), PGL(2, 29), PGL(2, 59), PGL(2, 61) or Aut(PSp(4, 4)) with |V (ΓOt (A) )| = 2 · 3 · 7, 2 · 3 · 11, 2 · 3 · 19, 2 · 7 · 41, 2 · 7 · 29, 2 · 29 · 59, 2 · 31 · 61 or 2 · 5 · 17, respectively. Let Aut(ΓOt (A) ) = PGL(2, 11), PGL(2, 19), PSL(2, 41), PGL(2, 29), PGL(2, 59) or PGL(2, 61). Then |V (ΓOt (A) )| = 2 · 3 · 11, 2 · 3 · 19, 2 · 7 · 41, 2 · 7 · 29, 2 · 29 · 59 or 2 · 31 · 61. Since A/Ot (A) is arc-transitive on ΓOt (A) , we have A/Ot (A)  A5 , and since A/OA (t ) ≤ Aut(ΓOt (A) ) is insolvable, by Proposition 2.7, A/Ot (A) has a normal subgroup B/Ot (A) such that B/Ot (A) = PSL(2, 11), PSL(2, 19), PSL(2, 41), PSL(2, 29), PSL(2, 59) or PSL(2, 61). Note that Mult(B/Ot (A)) = Z2 (see Table 2) and Ot (A) = Zt has odd order. By Lemma 2.8, B′ ∩ Ot (A) = 1 and B = B′ × Ot (A), implying that B′ ∼ = B/Ot (t ) is a non-abelian simple group. Since B/Ot (A) E A/Ot (A), we have B E A and since B′ is characteristic in B, we have B′ E A. This means that A has a non-abelian simple normal subgroup B′ , contrary to the hypothesis. Let Aut(ΓOt (A) ) = Aut(PSL(3, 4)). Then V (ΓOt (A) ) = 2 · 3 · 7 and |V (Γ )| = 2 · 3 · 7 · t, implying that (t , 6) =  1 because |V (Γ )| is square-free. Since A/Ot (A) is arc-transitive on ΓOt (A) , we have 2 · 3 · 5 · 7|A/Ot (A)|. Note that subgroup PSL(3, 4). Then A/Ot (A) ∩ PSL(3, 4) E A/Ot (A) and Aut(ΓOt (A) ) = Aut(PSL  (3, 4)) = PSL(3, 4).D6 has a normal  |(A/Ot (A))PSL(3, 4)||Aut(PSL(3, 4))| implies that 5 · 7|A/Ot (A) ∩ PSL(3, 4)|. Since the order of each maximal subgroup of PSL(3, 4) cannot be divisible by 5 · 7 by Atlas [4, pp. 23], we have A/Ot (A) ∩ PSL(3, 4) = PSL(3, 4), that is, A/Ot (A) has a normal subgroup B/Ot (A) such that B/Ot (A) = PSL(3, 4). Since Mult(PSL(3, 4)) = Z4 × Z12 (see Table 2) and (t , 6) = 1, by Lemma 2.8, B = B′ × Ot (A) = PSL(3, 4) × Zt . Since B E A and B′ = PSL(3, 4) is characteristic in B, we have B′ E A, contrary to the hypothesis. Let Aut(ΓOt (A) ) = Aut(PSp(4, 4)) = PSp(4, 4).Z4 . Then V (ΓOt (A) ) = 2 · 5 · 17. By a similar argument to the above paragraph, 2 · 52 · 17|A/Ot (A)| and 52 · 17|A/Ot (A) ∩ PSp(4, 4)|, and thus A/Ot (A) ∩ PSp(4, 4) = PSp(4, 4) by Atlas [4, pp. 44], that is, A/Ot (A) has a normal subgroup B/Ot (A) = PSp(4, 4). Note that Mult(PSp(4, 4)) = 1 (see Table 2). Again by Lemma 2.8, B = B′ × Ot (A) = PSp(4, 4) × Zt . Hence, B′ = PSp(4, 4) E A, contrary to the hypothesis. Now, A is solvable. Let F be the Fitting subgroup of A. Then F = Op (A) × Oq (A) × Or (A) ̸= 1, and F is cyclic because Os (A) = 1 or Zs for s = p, q, r. This implies that F ≤ CA (F ). On the other hand, CA (F ) ≤ F by Proposition 2.6, and thus CA (F ) = F . Suppose Fv ̸= 1 for some v ∈ V (Γ ). By Proposition 2.3, F has at most two orbits on V (Γ ) and since F is abelian, F has exactly two orbits, say ∆1 and ∆2 . Thus, Γ is a bipartite graph with bipartite sets ∆1 and ∆2 . Without loss of generality, we may assume that v ∈ ∆1 . Since F is abelian, Fv fixes every vertex in ∆1 and all orbits of Fv in ∆2 have same length. As Γ has valency 5 and is connected, we have Γ ∼ = Av F /F . = K5,5 , contrary to the fact that |V (Γ )| = 2pqr. Hence, Fv = 1 and Av ∼ Since F is cyclic, Aut(F ) is abelian, and since A/F = NA (F )/CA (F ) . Aut(F ), we have Av ∼ = Av F /F ≤ A/F is abelian. By Proposition 2.1, Av ∼ = Z5 , that is, Γ is one-regular.  Lemma 4.2. Assume that A has an insolvable minimal normal subgroup. Then A ∼ = PSL(2, 25), J1 , PSL(2, p) or PGL(2, p) with a prime p ≥ 29. Proof. Let N be an insolvable minimal normal subgroup of A. Then N ∼ = T s for some positive integer s and a non-abelian simple group T . As N is insolvable and |V (Γ )| = 2pqr, N is not semiregular. It follows that N has at most two orbits on V (Γ ) from Proposition 2.3. Let v N be an orbit of N on V (Γ ) containing the vertex v . Then pqr |v N |. Since N E A and Nv E Av ,







5|Nv |, implying that 5pqr |N |, and since |A|210 · 32 · 5 · r · q · p, we have |N | = 2i · 3j · 5 · r · q · p with 1 ≤ i ≤ 10 and 0 ≤ j ≤ 2. By Lemma 2.9, T is isomorphic to one group listed in Table 2. Since 3 ≤ r < q < p, |T | has at least four prime divisors, implying that T  A5 or A6 , and since p2 - |N |, we have N = T . Let π (N ) be the set  of all prime divisors of |N |. Note that 4 ≤ |π (N )| ≤ 6 and {2, 5, r , q, p} ⊆ π (N ). If 5 ∈ {r , q, p}, then 52 |N | because 5pqr |N |, and thus N ∼ = PSL(2, 25), PSp(4, 4), PSU(3, 4), Sz (32), J2 or PSL(2, p) with p ≥ 29 from Table 2. If 5 ̸∈ {r , q, p}, then |π (N )| ≥ 5. Again by Table 2, we have N ∼ = PSL(5, 2), PSL(2, 26 ), PSL(2, 28 ), M22 , M23 , J1 or PSL(2, p) with p ≥ 29. All these possible simple groups are listed in Table 3, and for each simple group N in Table 3, (r , q, p) can be easily determined from the order of N, except for the group PSL(2, p). Noting that N has at most two orbits on V (Γ ), we have |Nv | = |N |/2pqr or |N |/pqr. In particular, if N is transitive on V (Γ ), then N is arc-transitive on Γ because 5|Nv |. Let C = CA (N ). Since N is simple, C ∩ N = 1 and CN = C × N E A. Since A/CN  . Out(N ), we have A = (C × N ).O with O . Out(N ), where Out(N ) is the outer automorphism group of N. Since |A|210 · 32 · 5 · r · q · p and 5pqr |N |, C is a {2, 3}-group, and since |V (Γ )| = 2pqr, C has at least three orbits on V (Γ ). By Proposition 2.3, C is semiregular on V (Γ )

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D.-W. Yang et al. / Discrete Mathematics 339 (2016) 522–532 Table 3 Some possible simple groups N. Row

N

|N |

(r , q , p )

|Nv |

1 2 3 4 5 6 7 8 9 10 11 12

PSL(2, 25) PSp(4, 4) PSU(3, 4) Sz (32) J2 PSL(5, 2) PSL(2, 26 ) PSL(2, 28 ) M22 M23 J1 PSL(2, p)

23 · 3 · 52 · 13 28 · 32 · 52 · 17 26 · 3 · 52 · 13 210 · 52 · 31 · 41 27 · 33 · 52 · 7 210 · 32 · 5 · 7 · 31 26 · 32 · 5 · 7 · 13 28 · 3 · 5 · 17 · 257 27 · 32 · 5 · 7 · 11 27 · 32 · 5 · 7 · 11 · 23 23 · 3 · 5 · 7 · 11 · 19 p ≥ 29, a prime

(3, 5, 13) (3, 5, 17) (3, 5, 13) (5, 31, 41) (3, 5, 7) (3, 7, 31) (3, 7, 13) (3, 17, 257) (3, 7, 11) (7, 11, 23) (7, 11, 19)

22 27 25 29 26 29 25 27 26 26 22

·5 ·3·5 ·5 ·5 · 32 · 5 ·3·5 ·3·5 ·5 ·3·5 · 32 · 5 ·3·5

23 · 5 28 · 3 · 5 26 · 5 210 · 5 27 · 32 · 5 210 · 3 · 5 26 · 3 · 5 28 · 5 27 · 3 · 5 27 · 32 · 5 23 · 3 · 5

and ΓC is a connected pentavalent graph of order 2pqr /|C |, yielding that 2 - |C |. This means that C is a 3-group, and since  |V (Γ )| = 2pqr and |C ||V (Γ )|, either C = 1 or C ∼ = Z3 . In particular, if C ∼ = Z3 , then r = 3. First let N ∼ PSL ( 2 , 25 ) . Then | V ( Γ )| = 2 · 3 · 5 · 13 and |Nv | = 20 or 40 (see Table 3). Since PSL(2, 25) has no subgroups = of order 40 by Atlas [4, pp. 16], |Nv | = 20, and thus N is arc-transitive. By Example 3.1, Γ ∼ = G390 and A ∼ = PSL(2, 25). Let N = J1 . Then |V (Γ )| = 2 · 7 · 11 · 19. It follows that r = 7 and C = 1. By Atlas [4, pp. 36], Out(N ) = 1, and hence A = C × N = N.    Let N ∼ = PSL(2, p) with p ≥ 29. Suppose C ∼ = Z3 . Then r = 3 and 3|N | because pqr |N |. Since C × N E A, we have 32 |A|, and since |A : Av | = 2pqr is square-free, 3|Av |. By Proposition 2.1, Av is insolvable. Since A = (C × N ).O with O . Out(N ) and Out(N ) ∼ = Z2 , either A = N × C or A ∼ = (N × C ).Z2 , which implies that |Av : Nv | = 3 or 6 because N has at most two orbits. Since Av /Nv is solvable, Nv is insolvable, and by Proposition 2.7, Nv ∼ = A5 . Hence, |Av | = 180 or 360, which is impossible by Proposition 2.1. Hence, C = 1 and A ∼ = PSL(2, p) or PGL(2, p). To finish the proof, we suppose that N is isomorphic to some other group listed in Table 3 and aim to find a contradiction. Recall that N has at most two orbits on V (Γ ).



Case 1: N has exactly two orbits on V (Γ ). Let ∆1 and ∆2 be the two orbits of N. Then Γ is a bipartite graph with ∆1 and ∆2 as its bipartite sets, and A has a 2-element, say g, interchanging the two bipartite sets of Γ . Let N ∼ = M23 . Then |V (Γ )| = 2 · 7 · 11 · 23, r = 7 and C = 1. By Atlas [4, pp. 71], Out(N ) = 1, and A = (C × N ).O = N, contradicting the fact that A is transitive. Let N ∼ = PSp(4, 4), PSU(3, 4), PSL(2, 28 ), Sz (32) or PSL(5, 2). Note that H = N ⟨g ⟩ is arc-transitive and |Hv | = 2i |Nv | for some i ≥ 0. For N ∼ = PSp(4, 4), we have |Nv | = 28 · 3 · 5, and thus |Hv | = 2i |Nv | = 28+i · 3 · 5, which is impossible by Proposition 2.1. Similarly, for N ∼ = PSU(3, 4), PSL(2, 28 ), Sz (32) or PSL(5, 2), by Table 3 we have |Nv | = 26 · 5, 28 · 5, 210 · 5, 210 · 3 · 5, and |Hv | = 26+i · 5, 28+i · 5, 210+i · 5, 210+i · 3 · 5 respectively, a contradiction. Let N ∼ = J2 or PSL(2, 26 ). Then |Nv | = 27 · 32 · 5 or 26 · 3 · 5, respectively. By Atlas [4, pp. 42] and MAGMA [1], J2 or 6 PSL(2, 2 ) has no such subgroup Nv , a contradiction. Let N ∼ = M22 . Then |Nv | = 27 · 3 · 5. By Proposition 2.1, Nv = Hv ∼ = A6 L(2, 4). By MAGMA [1], Nv is a maximal subgroup of N and all such subgroups of N are conjugate. This implies that the two actions of N on ∆1 and ∆2 are equivalent. We may assume that v ∈ ∆1 , and the equivalence implies that Nv fixes a vertex in ∆2 . By Atlas [4, pp. 39], M22 has rank 3 with subdegrees 1, 16, 60, and hence Γ has valency at least 16, a contradiction. Case 2: N is transitive on V (Γ ). In this case, N is arc-transitive on Γ and by Proposition 2.1, |Nv | ̸= 25 · 5, 29 · 5, 29 · 3 · 5, 25 · 3 · 5 or 27 · 5, implying that N  PSU(3, 4), Sz (32), PSL(5, 2), PSL(2, 26 ) or PSL(2, 28 ). Let N ∼ = PSp(4, 4) or J2 . By Table 3, we have |Nv | = 27 · 3 · 5 or 26 · 32 · 5, respectively. Again by Proposition 2.1, we have Nv ∼ = A6 L(2, 4), S4 × S5 or AGL(2, 4), respectively. However, by MAGMA [1], PSp(4, 4) has no subgroup isomorphic to A6 L(2, 4) and by Atlas [4, pp. 42], J2 has no subgroup of order 26 · 32 · 5, a contradiction. Let N ∼ = M22 or M23 . Note that M23 has no subgroup isomorphic to S4 × S5 by Atlas [4, pp. 71]. By Table 3, |Nv | = 26 · 3 · 5 6 2 or 2 · 3 · 5, and by Proposition 2.1, Nv ∼ = ASL(2, 4) or AGL(2, 4), respectively. By MAGMA [1], all subgroups of M22 isomorphic to ASL(2, 4) have two conjugate classes, and all the subgroups of M23 isomorphic to AGL(2, 4) are conjugate. For N ∼ = M22 , let H1 ∼ = H2 ∼ = ASL(2, 4) be two subgroups of M22 which are not conjugate to each other. By Proposition 2.4, g Γ ∼ = Cos(N , Hi , Hi gi Hi ) for some 2-element gi ∈ N such that gi2 ∈ Hi , ⟨Hi , gi ⟩ = N and |Hi : Hi ∩ Hi i | = 5, where i = 1 or g 2 2. However, for each i = 1 or 2, there is no such a 2-element gi such that gi ∈ Hi , ⟨Hi , gi ⟩ = N and |Hi : Hi ∩ Hi i | = 5 by ∼ ∼ MAGMA [1], a contradiction. Similarly, for N ∼ , Γ M Cos ( N , H , HgH ) for some element g, where H = N AGL (2, 4). = 23 = v = Again by MAGMA [1], there is also no element g such that g 2 ∈ H, ⟨H , g ⟩ = N and |H : H ∩ H g | = 5, a contradiction.  Now, we are ready to prove Theorem 1.1.

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The proof of Theorem 1.1. By Lemmas 4.1 and 4.2, either Γ is one-regular, or A ∼ = PSL(2, 25), J1 , PSL(2, p) or PGL(2, p) with a prime p ≥ 29.   Assume that Γ is one-regular. By Proposition 2.5, we have Γ ∼ = CD ℓpqr , where 5(p − 1), 5(q − 1) and either r = 5 or 5(r − 1). If r = 5, then there are exactly 4 non-isomorphic such graphs for a given order, while there are 16 non-isomorphic



such graphs if 5(r − 1). Assume that A ∼ = PSL(2, 25) or J1 . Then |V (Γ )| = 2 · 3 · 5 · 13 or 2 · 7 · 11 · 19, respectively. By Example 3.1 and Lemma 3.3, we have Γ ∼ = G390 or G2926 .  Assume that A ∼ = PSL(2, p) or PGL(2, p) with p ≥ 29. Since |A : Av | = 2pqr and A is insolvable, 2|Av |, and by Propositions 2.1 and 2.7, Av ∼ = D5 , D10 or A5 . Let Av ∼ = D5 . Then |A| = 22 · 5 · r · q · p and 23 - |A|. It follows that A ∼ = PSL(2, p). Since |PSL(2, p)| = p(p−12)(p+1) , we have p(p − 1)(p + 1) = 23 · 5 · r · q · p, and thus (p + 1)(p − 1) = 23 · 5 · r · q. Clearly, either p + 1 or p − 1 has a divisor 3, which p+1 p−1 implies that 23 · 5 · r · q has a divisor 3, yielding that r = 3. Note that ( 2 , 2 ) = 1 and (p + 1)(p − 1) = 23 · 3 · 5 · q with



q > 3. If q(p + 1), then (p − 1)23 · 3 · 5, that is, p − 1 ∈ {2 · 3, 22 · 3, 23 · 3, 2 · 5, 22 · 5, 23 · 5, 2 · 3 · 5, 22 · 3 · 5, 23 · 3 · 5}. It follows that p = 7, 11, 13, 31, 41 or 61. Since p ≥ 29 and A ∼ = PSL(2, p) has order 22 · 3 · 5 · q · p, we have p = 61. Similarly,  (i) 3   if q (p − 1), then (p + 1) 2 · 3 · 5 and p = 29 or 59. By Examples 3.4, 3.7 and 3.8 and Lemma 3.5, Γ ∼ = G1218 with i = 1 or



(j)



(j)

2, G10,266 or G11,346 with 1 ≤ j ≤ 6.

p(p−1)(p+1)

or p(p − 1)(p + 1). Let Av ∼ = D10 . Then |A| = 23 · 5 · r · q · p. Since A ∼ = PSL(2, p) or PGL(2, p), we have |A| = 2 ∼ Hence, p(p − 1)(p + 1) = 2k · 5 · r · q · p, where k = 4 for A ∼ PSL ( 2 , p ) and k = 3 for A PGL ( 2 , p ) . By a similar argument = =  as the previous paragraph, r = 3 and (p + 1)(p − 1) = 2k · 3 · 5 · q. For q(p + 1), either p = 41 and A ∼ = PSL(2, 41), or p = 61 and A ∼ = PGL(2, 61). For q(p − 1), p = 29 or 59 and A ∼ = PGL(2, p). Let A ∼ = PSL(2, 41), PGL(2, 59) or PSL(2, 61).



(7)

(7)

By Examples 3.6–3.8, Γ ∼ = G1722 , G10,266 or G11,346 . Now let A ∼ = PGL(2, 29). Then A has a normal subgroup B ∼ = PSL(2, 29). If B has two orbits, then Bv = Av ∼ = D10 by Proposition 2.1, which is impossible by Proposition 2.7. It follows that B is (3) arc-transitive, and by Example 3.4 and Lemma 3.5, Γ ∼ = G1218 . Let Av ∼ = A5 . By Proposition 2.7, 5(p ± 1). Since |A : Av | = 2pqr, |A| is divisible by 8, but not by 16. Since |A| =

|PSL(2, p)| = p(p−12)(p+1) or |PGL(2, p)| = p(p − 1)(p + 1), we have p ≡ ±7 (mod 16) for A ∼ = PSL(2, p) or p ≡ ±3 (mod 8) for A ∼ = PGL(2, p). Since 5(p ± 1), we have p ≡ 9, 39, 41, 71(mod 80) for A ∼ = PSL(2, p) or p ≡ 11, 19, 21, 29(mod 40) for A∼ = PGL(2, p). By Lemmas 3.10 and 3.12, Γ ∼ = GPSL(2,p) or GPGL(2,p) . 

Acknowledgment This work was supported by the National Natural Science Foundation of China (11571035, 11231008, 11171020, 11301159, 11371052). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]

W. Bosma, C. Cannon, C. Playoust, The MAGMA algebra system I: The user language, J. Symbolic Comput. 24 (1997) 235–265. P.J. Cameron, G.R. Omidi, B. Tayfeh-Rezaie, 3-Designs from PGL(2, q), Electron. J. Combin. 13 (2006) 1–11. Y. Cheng, J. Oxley, On weakly symmetric graphs of order twice a prime, J. Combin. Theory Ser. B (42) (1987) 196–211. J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker, R.A Wilson, Atlas of Finite Group, Clarendon Press, Oxford, 1985. L.E. Dickson, Linear Groups and Exposition of the Galois Field Theory, Dover, 1958. D.Ž. Djoković, A class of finite group-amalgams, Proc. Amer. Math. Soc. 80 (1980) 22–26. D.Ž. Djoković, G.L. Miller, Regular groups of automorphisms of cubic graphs, J. Combin. Theory Ser. B 29 (1980) 195–230. X.G. Fang, C.E. Praeger, Finite two-arc transitive graphs admitting a Suzuki simple group, Comm. Algebra 27 (1999) 3727–3754. Y.-Q. Feng, M. Ghasemi, D.-W. Yang, Cubic symmetric graphs of order 8p3 , Discrete Math. 318 (2014) 62–70. Y.-Q. Feng, J.H. Kwak, Cubic symmetric graphs of order a small number times a prime or a prime square, J. Combin. Theory Ser. B 97 (2007) 627–646. Y.-Q. Feng, Y.-T. Li, One-regular graphs of square-free order of prime valency, European J. Combin. 32 (2011) 265–275. C.D. Godsil, On the full automorphism group of a graph, Combinatorica 1 (1981) 243–256. D. Gorenstein, Finite Simple Groups, Pleenum Press, New York, 1982. S.-T. Guo, Y.-Q. Feng, A note on pentavalent s-transitive graphs, Discrete Math. 312 (2012) 2214–2216. S.-T. Guo, J.-X. Zhou, Y.-Q. Feng, Pentavalent symmetric graphs of order 12p, Electron. J. Combin. 18 (2011) #P233. X.-H. Hua, Y.-Q. Feng, Pentavalent symmetric graphs of order 8p, J. Beijing Jiaotong University 35 (2011) 132–135, 141. X.-H. Hua, Y.-Q. Feng, J. Lee, Pentavalent symmetric graphs of order 2pq, Discrete Math. 311 (2011) 2259–2267. B. Huppert, Eudiche Gruppen I, Springer-Verlag, 1967. J.H. Kwak, Y.S. Kwon, J.M. Oh, Infinitely many one-regular Cayley graphs on dihedral groups of any prescribed valency, J. Combin. Theory Ser. B 98 (2008) 585–598. C.H. Li, Z.P. Lu, G.X. Wang, The vertex-transitive and edge-transitive tetravalent graphs of square-free order, J. Algebraic Combin. 42 (2015) 25–50. B. Ling, C.X. Wu, B.G. Lou, Pentavalent symmetric graphs of order 30p, Bull. Aust. Math. Soc. 90 (2014) 353–362. P. Lorimer, Vertex-transitive graphs: symmetric graphs of prime valency, J. Graph Theory 8 (1984) 55–68. L. Morgan, On symmetric and locally finite actions of groups on the quintic tree, Discrete Math. 313 (2013) 2486–2492. J.M. Oh, A classification of cubic s-regular graphs of order 14p, Discrete Math. 309 (2009) 2721–2726. J.M. Oh, A classification of cubic s-regular graphs of order 16p, Discrete Math. 309 (2009) 3150–3155. J.M. Pan, B.G. Lou, C.F. Liu, Arc-transitive pentavalent graphs of order 4pq, Electron. J. Combin. 20 (2013) #P36. P. Potočnik, A list of 4-valent 2-arc-transitive graphs and finite faithful amalgams of index (4, 2), European J. Combin. (2009) 1323–1336.

532

D.-W. Yang et al. / Discrete Mathematics 339 (2016) 522–532

[28] P. Potočnik, P. Spiga, G. Verret, Bounding the order of the vertex-stabiliser in 3-valent vertex-transitive and 4-valent arc-transitive graphs, J. Combin. Theory Ser. B 111 (2015) 148–180. [29] J. Schur, Über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen, J. Reine Angew. Math. 127 (1904) 20–50. [30] R. Weiss, Presentation for (G, s)-transitive graphs of small valency, Math. Proc. Cambridge Philos. Soc. 101 (1987) 7–20. [31] R.A. Wilson, The Finite Simple Groups, in: Graduate Texts in Mathematics, vol. 251, Springer-Verlag, London, 2009. [32] M.Y. Xu, Automorphism groups and isomorphisms of Cayley digraphs, Discrete Math. 182 (1998) 309–319. [33] J.-X. Zhou, Y.-Q. Feng, Tetravalent s-transitive graphs of order twice a prime power, J. Aust. Math. Soc. 88 (2010) 277–288.