ARS MATHEMATICA CONTEMPORANEA
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 15 (2018) 53-65
https://doi.org/10.26493/1855-3974.1161.3b9
(Also available at http://amc-journal.eu)
On prime-valent symmetric graphs of square-free order*

Jiangmin Pan t
School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan, P. R. China
Bo Ling
School ofMathematics and Computer Science, Yunnan Minzu University, Kunming, Yunnan, P. R. China
Suyun Ding
School ofMathematics and Statistics, Yunnan University, Kunming, Yunnan, P. R. China
Received 16 July 2016, accepted 27 May 2017, published online 10 November 2017
Symmetric graphs of valencies 3, 4 and 5 and square-free order have been classified in the literature. In this paper, we will present a complete classification of symmetric graphs of square-free order and any prime valency which admit a soluble arc-transitive group, and a complete classification of 7-valent symmetric graphs of square-free order.
Keywords: Symmetric graph, normal quotient graph, automorphism group. Math. Subj. Class.: 20B15, 20B30, 05C25
1 Introduction
Throughout the paper, graphs considered are assumed to be undirected and simple with valency at least three.
For a graph r, denote by Vr and Ar the vertex set and arc set of r respectively, denote by Autr the full automorphism group of r, and denote by r (a) the set of neighbors of a
*The authors are very grateful to the referees for their valuable comments. This work was partially supported by the National Natural Science Foundation of China (11461007, 11231008). t Corresponding author.
E-mail addresses: jmpan@ynu.edu.cn (Jiangmin Pan), bolinggxu@163.com (Bo Ling), 1328897542@qq.com (Suyun Ding)
Abstract
©® This work is licensed under https://creativecommons.org/licenses/by/3.0/
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Ars Math. Contemp. 15 (2018) 39-51
vertex a in r. Then r is called X-vertex-transitive or X-arc-transitive, with X < Autr, if X is transitive on Vr or Ar respectively. An arc-transitive graph is also called a symmetric graph. In particular, r is called arc-regular if Autr is regular on Ar.
For a positive integer s, an s-arc of a graph r is a sequence v0, v\,..., vs of s + 1 vertices such that vj_i, vj are adjacent for 1 < i < s and vj_i = vi+1 for 1 < i < s - 1. If r has an s-arc and X < Autr is transitive on the set of s-arcs of r, then r is called (X, s)-arc-transitive. If r is (Autr, s)-arc-transitivebutnot (Autr, s + 1)-arc-transitive, then r is simply called s-transitive.
Characterizing symmetric graphs was initiated by a nice result of Tutte (1949) which says that there exists no s-arc-transitive cubic graph with s > 6. This result was generalized by Weiss [27] who proved that there is no s-arc-transitive graph with s > 8 of valency at least 3. Since then, studying transitive graphs has been one of the main topics in algebraic graph theory, and numerous results have been obtained. In particular, transitive graphs of square-free order (not divisible by the square of a prime) have received considerable attention; for example, symmetric graphs of valencies 3, 4 and 5 and square-free order have been classified by [16, 17] and [6] respectively, and arc-regular graphs of square-free order and prime valency have been determined by [9]. The main purpose of this paper is to give a complete classification of symmetric graphs of square-free order and prime valency admitting a soluble arc-transitive group, and a complete classification of 7-valent symmetric graphs of square-free order.
The terminology and notation used in this paper are standard. For example, we denote by J1 the Janko simple group, by HS the Higman-Sims simple group, and by Mn, with n = 11,12,22, 23, 24, the five Mathieu simple groups. For a positive integer m, denote by Am and Sm the alternating group and symmetric group of degree m, and by Zm, Fm and Dm (with m even) the cyclic group, Frobenius group and dihedral group of order m respectively. Given two groups N and H, denote by N x 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.
A graph r is called a Cayley graph if there exists a group G and a subset S C G \ {1} with S = S-1: = {s-1 | s € S} such that the vertex set V r = G andavertex x is adjacent to a vertex y if and only if yx-1 € S. This Cayley graph is denoted by Cay(G, S). The following Cayley graphs of dihedral groups give rise to an infinite family of prime-valent symmetric graphs, where the first two letters 'CD' of the name of the graph CD2m,p,k stand for 'Cayley graph of a dihedral group'.
Example 1.1. Let G = (a, b | am = b2 = 1, ab = a-1} = D2m with m a positive integer, and let p be an odd prime and k a solution of the equation
xp-1 + xp-2 +-----+ x +1 = 0 (mod m).
Set
CD2m,p,fc = Cay(G, {b, ab, afc+1b,..., afcP-2+fcP-3+-+fc+1b}).
The following theorem determines the prime-valent symmetric graphs of square-free order which admit a soluble arc-transitive automorphism group. We remark that cubic graphs which admit a soluble edge-transitive or arc-transitive automorphism group have been characterized by [20] and [8], respectively.
J. Pan et al.: On prime-valent symmetric graphs of square-free order
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Theorem 1.2. Let r be a connected p-valent symmetric graph of square-free order n with p an odd prime, and suppose that r admits a soluble arc-transitive automorphism group G. Then either
(1)	r = Kpp, and G = (((Zp : Ztl) x (Zp : Z¡2)).Zr).Z2 < Sp I Z2, where lir | p — 1 for i = 1, 2; or
(2)	r = CDn p k, n = 2 • psp1p2 • • • pt and G = Autr = Dn:Zp, where 0 < s < 1, t > 1, and p1,p2,... ,pt are distinct primes such that p | pi — 1 for i = 1, 2,... ,t. Further, there are exactly (p — 1)t-1 non-isomorphic such graphs of order n.
The next theorem present a complete classification of 7-valent symmetric graphs of square-free order, where the graph C330 in Table 1 is introduced in Example 3.2 for convenience.
Theorem 1.3. Let r be a connected 7-valent symmetric graph of square-free order n. Then one of the following statements holds.
(1)	r = CD„i7jk, and the tuple (n, Autr) is as in part (2) of Theorem 1.2 with p = 7.
(2)	The triple (r, n, Autr) lies in Table 1.
(3)	Autr = PSL(2,p) or PGL(2,p), where p > 13 is a prime such that p(p2 — 1) | 225 • 34 • 52 • 7n.
Table 1: Two 'sporadic' 7-valent symmetric graphs Row r n Autr (Autr )a Transitivity Remark
1	K7 7 14 S71Z2 S7.S6 3-transitive bipartite
2	C330 330 M22.Z2 Z4 : SL(3,2) 2-transitive not bipartite
Remark 1.4. Graphs appearing in part (3) of Theorem 1.3 can be expressed as coset graphs of PSL(2,p) or PGL(2,p) (refer to [10] for the definition of the coset graph). However, it seems infeasible to determine all the possible values of p (and so the corresponding symmetric graphs r) for general square-free integer n.
2 Preliminaries
In this section, we introduce some preliminary results that will be used later.
For a group G with a subgroup H, let CG(H) and NG(H) denote the centralizer and normalizer of H in G, respectively.
Lemma 2.1 ([14, Ch. I, Lemma 4.5]). Let G be a group and H a subgroup of G. Then
Ng(H)/Cg(H) < Aut(H).
For a group G, the largest nilpotent normal subgroup of G is called the Fitting subgroup of G. Clearly, the Fitting subgroup is a characteristic subgroup. The next lemma gives a property of the Fitting subgroup of soluble groups.
Lemma 2.2 ([26, P. 30, Corollary]). Let F be the Fitting subgroup of a soluble group G. Then F = 1 and CG(F) < F.
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The maximal subgroups of the simple group PSL(2, q) are known, see [5, Section 239].
Lemma 2.3. Let T = PSL(2, q), where q = pn > 5 with p a prime. Then a maximal subgroup of T is isomorphic to one of the following groups, where d = (2,p — 1).
(1)	D 2(q—i), where q = 5,7,9,11;
d
(2)	D 2(q+i), where q = 7, 9;
d
(3)	Zjn : Z q—i;
p d
(4)	A4, where q = 5, or q = p = 3,13, 27, 37 (mod 40);
(5)	S4, where q = p = ±1 (mod 8);
(6)	A5, where q = p = ±1 (mod 5), or q = p2 = —1 (mod 5) with p an odd prime;
(7)	PSL(2, r), where q = rm with m an odd prime;
(8)	PGL(2,r), where q = r2.
By [2, Theorem 2], one may easily derive the maximal subgroups of PGL(2,p).
Lemma 2.4. Let T = PGL(2,p) with p > 5 a prime. Then a maximal subgroup of T is isomorphic to one of the following groups:
(1)	Zp : Zp_i;
(2)	D2(p+1);
(3)	D2(p_i), wherep > 7;
(4)	S4, where p = ±3 (mod 8);
(5)	PSL(2,p).
A group G is called perfect if G = G', the commutator subgroup; and an extension G = N.H is called a central extension if N C Z(G), the center of G. If a group G is perfect and G/Z(G) is isomorphic to a simple group T, then G is called a covering group of T. Schur [25] showed that a simple (and, more generally, perfect) group T possesses a universal covering group G with the property that every covering group of T is a homomor-phic image of G, in this case, the center Z(G) is called the Schur multiplier of T, denoted by Mult(T), see [12, P. 43]. The Schur multipliers of nonabelian simple groups are known (see [12, P. 302]), and the following lemma is easy to prove (see [23, Lemma 2.11]).
Lemma 2.5. Let G = N.T, where N is a cyclic group and T is a nonabelian simple group. Then G = N.T is a central extension. Further, G = NG' and G' = M.T, where M is contained in G' n N and is isomorphic to a subgroup of Mult(T).
The following lemma characterizes the vertex stabilizers of 7-valent symmetric graphs, see [13, Theorem 1.1].
Lemma 2.6. Let r be a connected 7-valent (X, s)-arc-transitive graph, where X < Autr and s > 1. Then one of the following holds, where a e Vr.
(1) If Xa is soluble, then s < 3 and \Xa\ | 252. Further, the couple (s, Xa) is listed in Table 2.
J. Pan et al.: On prime-valent symmetric graphs of square-free order	57
Table 2: Soluble vertex-stabilizers of 7-valent-symmetric graphs.
s	1	2	3
Xa	Z7, D14, F21, D14 X Z2,	AGL(1,7), AGL(1, 7) x Z2,	AGL(1, 7) x Z6
	F21 X Z3	AGL(1, 7) x Z3	
Table 3: Insoluble vertex-stabilizers of 7-valent symmetric graphs.
s	2	3
Xa	PSL(3, 2), ASL(3, 2),	PSL(3, 2)xS4, A7xA6,
	ASL(3, 2)xZ2, A7, S7	S7xS6, (A7xA6):Z2, Z2:(SL(2, 2)xSL(3, 2)), Z20:(SL(2, 2)xSL(3, 2))
|Xa|		26^32^7, 26-34-52-7,
		28^34^52^7, 27^34^52^7, 210^32^7, 224^32^7
(2) If Xa is insoluble, then \Xa\ | 224 • 34 • 52 • 7. Further, the couple (s,Xa) lies in Table 3.
Analyzing a graph in terms of its normal quotients is a typical method for studying vertex-transitive graphs. Let r be an X-vertex-transitive graph with X < Autr, and suppose that X has a normal subgroup N which is intransitive on V r. Denote by VrN the set of all N-orbits on Vr. Then the normal quotient graph rN of r induced by N is defined as the graph with vertex set VrN, and B is adjacent to C in rN if and only if there exist vertices fi G B and 7 G C such that fi is adjacent to 7 in r .In particular, if for any adjacent vertices B and C in VrN, the induced subgraph [B, C] = mK2 is a perfect matching, where m = \B\ = \C\, then r is called a regular cover (or normal cover) of rN.
The following theorem gives a basic reduction method for studying vertex-transitive locally primitive graphs (see [18, Lemma 2.5]), which slightly improves a nice result of Praeger [24, Theorem 4.1]. Recall that, a graph r is called X-locally primitive if the vertex stabilizer Xa acts primitively on the neighbour set r(a) for each a G Vr. Obviously, symmetric graphs with odd prime valency are locally primitive.
Theorem 2.7. Let r be an X-vertex-transitive locally primitive graph, where X < Autr, and let N < X have at least three orbits on Vr. Then the following statements hold.
(i)	N is semi-regular on Vr, X/N < AutrN, and r is a regular N-cover of rN;
(ii)	Xa = (X/N)Y, where a G Vr and 7 G VrN;
(iii)	r is (X, s)-arc-transitive if and only if rN is (X/N, s)-arc-transitive, where 1 < s < 5 or s = 7.
Symmetric graphs of prime-valency and order twice a prime are known, see [3].
Lemma 2.8. Let r be a connected symmetric graph of odd prime valency p and order 2r with r a prime. Then one of the following statements holds.
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Ars Math. Contemp. 15 (2018) 39-51
(1)	r = O2 and Autr = S5;
(2)	r = K2r with p = 2r — 1, and Autr = S2r;
(3)	r = Kr,r with p = r, and Autr = Sr I S2;
(4)	r = CD2r,p,k (which, up to isomorphism, is independent of the choice of k in this case), where p | r — 1, and one of the following statements holds.
(i)	(r,p) = (7, 3) and Autr = PGL(2, 7);
(ii)	(r,p) = (ll, 5) and Autr = PGL(2,11);
(iii)	(r,p) = (7,3) and (11,5), and Autr = D2r : Zp.
Lemma 2.9 ([19, Theorem 1.1]). Let r be a connected 7-valent symmetric graph of order 2pq with p > q odd primes. Then one of the following holds:
(1)	Autr = PSL(2,p) withp > 13;
(2)	q = 7 or 7 | p — 1, 7 | q — 1, and r = CD2pq,7,k (as in Example 1.1). 3 A lemma and an example
In this section, we give a technical lemma and introduce an example appearing in Theorem 1.3.
The following is an assertion regarding simple groups, its proof depends on the classification of simple groups, see [12, P. 134-136].
Lemma 3.1. Let m be an odd square-free integer with at least three prime factors, and let T be a nonabelian simple group such that 28m | |T| and |T| | 225 • 34 • 52 • 7m. Then the couple (T, |T |) is listed in Table 4, where p in part 4 is the largest prime factor of m and p > 13.
Proof. If T is a sporadic simple group, by [12, P. 135-136], T = M22, M23, M24, Ji, HS or Ru, as in part 1 of Table 4. If T = An is an alternating group, since 36 does not divide |T| and 36 | |Ai51, we have n < 14, it then easily follows that T = An, A12, A13 or A14, as in part 2 of Table 4.
Now, suppose that T is a simple group of Lie type defined on the re-elements field GF(re), where r is a prime. If T is of exceptional Lie type, by [12, P. 135], T = Sz(512) or 3D4(2), as listed in part 3 of Table 4. Consider the case where T is of classical Lie type. Since re | |T|, we have that e = 1 if r > 7, and e < 2 if r = 7, by [12, P. 135], which give rise to examples T = PSL(2,p) with p > 13 a prime (noting that PSL(2,p) with p = 5, 7,11 does not satisfy the hypothesis of Lemma 3.1) and T = PSL(2,49). If r = 5, as 54 / |T|, we conclude from [12, P. 135] that T = PSL(2,125). For the case where r < 3, since 36, 54 and 73 do not divide |T|, by [12, P. 135] and with the help of Magma [1], we conclude that T is isomorphic to one of the groups listed in part 4 of Table 4.	□
Given a permutation group G, a direct computation by Magma program [1] can determine all orbital graphs of G (see [7, P. 66] for the definition of orbital graph), or in other words, can determine all symmetric graphs which admit G as an arc-transitive automorphism group. It is then easy to have the following example.
Example 3.2. There is a unique connected 7-valent symmetric graph of order 330, denoted by C330, which admits M22 or M22 .Z2 as an arc-transitive automorphism group. The graph C330 satisfies the conditions in Row 2 of Table 1.
J. Pan et al.: On prime-valent symmetric graphs of square-free order	59
Table 4: Nonabelian simple groups T with 28m | |T| and |T| | 225 • 34 • 52 • 7m.
Part	T	|T |
1	M22	
	M23	2^32^74L23
	M24	210^33^74L23
	J1	23^5^1L19
	HS	29^32^53^741
	Ru	214^33^53^743^29
2	A11	27^52^11
	A12	29^35^52^741
	A13	29-35-52-7-11-13
	A14	210^35^52^724143
3	Sz(512)	218^743^73409
	3D4 (2)	212^34^7243
4	PSL (2,p)	p(p2 - 1)/2
	PSL 2, 49)	24^52^72
	PSL 2, 125)	22^32^53^31
	PSL( 2, 26)	26-32-5-7-13
	PSL( 2, 29)	29^33^749^73
PSL PSL PSL PSL PSL PSL PSL PSL PSL PSL PSL PSL PSL PSp PSp PSp ( PQ ( 7,4) PQ( 9, 2) PQ+ (10,2) PQ-( 8,2) PQ-( 8,4)
212) 215) 218) 221) 224)
8) 16) 64) 4)
212 32 5 7 13 17 241
215 218
221
224
->9
711 31151 331 5^74349^37^73409 7243427^337^5419 5-7-13-17-97-241-257-673
29-32-72-73 212-32-52-7-13-17
218 212
210
220
215
221 218
216 212
218
216
220
212
5^724349^73
52-7-17
5^31
527111731
5^72^31
5^72^31427
53^74347
52^747
5^7243
53^74347
52^747
52-7-17-31
5 7 17
224-34-53-7-13-17-257
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Ars Math. Contemp. 15 (2018) 39-51
4	Proof of Theorem 1.2
In this section, we prove Theorem 1.2 which in particular gives a partial proof of Theorem 1.3.
Let r be a connected symmetric graph of odd prime valency p and square-free order n, and let G be a soluble arc-transitive automorphism group of r. Since r is of odd valency, n is even. Set n = 2pip2 • • • pt, with pi, p2,..., pt distinct odd primes.
If t = 1, by Lemma 2.8, r = CD2piiPjfc (as in part (2) of Theorem 1.2) or r = Kp,p. If r = Kp,p, then G < Autr = Sp IZ2, as G is arc-transitive on r, Z^ < G and G^S^, it is then routine to show that G = (((Zp : Zll) x (Zp : Z;2 )).Zr).Z2, where /¿r | p - 1 for i = 1, 2, as in part (1) of Theorem 1.2.
Suppose t > 2 in the following. Let F be the Fitting subgroup of G. By Lemma 2.2, F = 1. As |Vr| = 2p1p2 • • • pt, G has no nontrivial normal Sylow a-subgroup, where a G {2,p1,p2,... ,pt} is aprime, hence
F = 02(G) x Opi(G) x---x Opt(G),
where 02(G) and 0pi(G) with i = 1,2,... ,t denote the largest normal 2- and p4-subgroups of G, respectively.
For each prime q G {2,p1,p2,... ,pt}, since t > 2, 0q(G) has at least six orbits on Vr, by Theorem 2.7, 0q(G) is semi-regular on Vr, so is F and 0q(G) < Zq. Hence F < Z„ is cyclic and CG(F) = F by Lemma 2.2.
If F is transitive on V r, then F = Zn is regular on V r and r is a Cayley graph of F. Set r = Cay(F, S), where S = S-1 C F \ {1} with size |S| = p. Since F < G, by [11, Lemma 2.9], G < F:Aut(F, S), so Gi < Aut(F, S) < Aut(F) is transitive on r(1) = S, where 1 denotes the vertex of r corresponding to the identity element of F, thus elements in S have the same order, say h. Clearly, h = 2 as F has a unique involution. If h > 2, as
5	= S-1, |S | is even, which is a contradiction.
If F has at least three orbits on Vr, then Theorem 2.7 implies that the normal quotient graph rF is G/F-arc-transitive; however, by Lemma 2.2, G/F = G/Cg(F) < Aut(F) is abelian, it forces that G/F is regular on VrF, and so G/F is not transitive on Ar, also yielding a contradiction.
Thus, F has exactly two orbits on Vr, and F = Zn. Because t > 2, F has a nontrivial normal subgroup K = Zp2p3...pt. Since K < G has 2p1 orbits on Vr, by Theorem 2.7, rK is a G/K-arc-transitive graph of valency p and order 2p1, and r is a regular K-cover of rK. Such covers have been classified by [21, Theorem 1.1], hence the triple (r, K, rK) (as (r, Zn, S) there) satisfies parts (1)-(5) of [21, Theorem 1.1]. Since |K| = 2, parts (1)-(3) are impossible. For part (4), since n is square-free, p1 / |K|, by [22, Theorem 1.1], r = CDnjPjfc. For part (5), noting that p1 / |K|, part (5)(ii) is not possible, we also have r = CDn p fc. Finally, the last statement in part (2) of Theorem 1.2 is true by [9, Theorem 3.1]. This completes the proof of Theorem 1.2.
5 Proof of Theorem 1.3
We will prove Theorem 1.3 in this final section.
Let r be a connected 7-valent symmetric graph of square-free order n. Since r is of odd valency, n is even, so we may write
n := 2m = 2p1p2 . . .pt,
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where... ,pt are distinct odd primes. Let A = Autr.
Lemma 5.1. If t < 2, then Theorem 1.3 is true.
Proof. If t =1, by Lemma 2.8, r = CD2pi,7,fc (as in part (1) of Theorem 1.3), or r = K7,7 (as in Row 1 of Table 1).
If t = 2, by Lemma 2.9, r = CD2piP2,7,fc (as in part (1) of Theorem 1.3), or A = PSL(2,p) or PGL(2,p) with p > 13 aprime, satisfying part (3) of Theorem 1.3.	□
Thus, assume t > 3 in the following, and assume inductively that Theorem 1.3 is true for the graph which satisfies assumption of Theorem 1.3 and is of order less than n. Let a € Vr. By Lemma 2.6, |Aa| | 224 • 34 • 52 • 7, hence |A| = |Aa||Vr| divides 225 • 34 • 52 • 7m. Let R be the soluble radical of A, that is, the largest soluble normal subgroup of A. Obviously, the soluble radical of A/R equals 1. The next lemma treats the case R =1.
Lemma 5.2. Suppose R =1 and t > 3. Then either Autr = PSL(2,p) or PGL(2,p) with p > 13 a prime such that p(p2 — 1) | 225 • 34 • 52 • 7n, as in part (2) of Theorem 1.3; or r = C330 and Autr = M22.2, as in Row 2 of Table 1.
Proof. Let N be a minimal normal subgroup of A, and let C = CA(N). Since R = 1, N = Td and |N | = |T |d divides 225 • 34 • 52 • 7m, where T is a nonabelian simple group and d > 1.
Claim 1. C =1.
Assume, on the contrary, C =1. Then C is insoluble as R = 1. If C is semi-regular on Vr, then |C| | n, so C is of square-free order and hence soluble, which is a contradiction. Thus Ca = 1. Since r is connected and C < A, we have 1 = C^(a) < A^(a), so 7 | |Ca|. Arguing similarly, one may have 7 | |Na|. Now, since N n C =1, (N, C} = N x C < A, so Na x Ca < Aa, hence 72 | |Aa|, which is a contradiction by Lemma 2.6. Therefore, C = 1.
Claim 2. A is almost simple and the tuple (T, |T |) is listed in Table 4.
As discussed above, 7 | |Na|. Then by Theorem 2.7, N has at most two orbits on Vr, hence m divides |N : Na|, we further conclude that 7m | |N|, 7 | |T| and m | |T|.
Without a loss of generality, let pt be the largest prime dividing n. As t > 3, pt > 7, and as md = (p1p2 • • • pt)d divides 225 • 34 • 52 • 7p1p2 • • • pt, we have d < 2. If d =2, the only possibility is t = 3 and m = 3 • 5 • 7, so |T|2 | 225 • 35 • 53 • 72, hence |T| | 212 • 32 • 5 • 7; recall that m | |T|, by [15, Theorem III], T = A; with l = 7 or 8, and N = A2. By Claim 1, C =1, then Lemma 2.1 implies A = A/C < Aut(N) = S; I Z2, and as N = A2 is a minimal normal subgroup of A, we conclude that A = A; I Z2, (A; I Z2).Z2 or S; I Z2. Since | Aa | = ^, a direct computation by Magma [1] shows that no graph r exists in this case, a contradiction. Thus, d =1 and N = T, and by Lemma 2.1, A < Aut(T) is almost simple. Recall that |T| divides 225 • 34 • 52 • 7m and 7m divides |T|, and noting that 4 | |T| as T is nonabelian simple, we have 28m | |T|. By Lemma 3.1, the couple (T, |T|) is listed in Table 4.
Now, we will analyse all the candidates of T in Table 4, thus proving Lemma 5.2. Recall that n = 2m and |T : Ta| = m or 2m. Denote by Out(T) the outer automorphism group of T.
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AssumeT = PSL(2,p) withp > 13 aprime. Thenp(p2 - 1) | 225 • 34 • 52 • 7n, andas Out(T) = Z2 (see [12, P. 135]), we have A = PSL(2,p) or PGL(2,p), the lemma is true.
Assume T = M22. Then m = 3 • 5 • 11 = 165 and n = 330. Since Out(M22) = Z2, A = M22 or M22.Z2. By Example 3.2, r = C330, satisfying the conditions in Row 2 of Table 1.
Assume T = M23. Then m = 3 • 11 • 23,5 • 11 • 23 or 3 • 5 • 11 • 23. Since Out(M23) = 1, A = T = M23 and so |T : Ta| = 2m = 1518,2530 or 7590. However, by [4], M23 has no subgroup with index 1518, 2530 or 7590, a contradiction.
Assume T = Ji. Then m = 627,1045 or 3135. Since Out(Ji) = 1, we have A = T and |T : Ta| = 2m = 1254,2090 or 6270. By [4], Ji has no subgroup with index 1254,2090 or 6270, which is a contradiction.
Suppose T = A12. Then m = 165 and |T : Ta| = 165 or 330. By [4], A12 has no subgroup with index 165 or 330, yielding a contradiction.
Suppose T = PSL(2,49). Then m = 105 and |T : Ta| = 105 or 210, it follows |Ta| = 560 or 280 respectively. By Lemma 2.3, PSL(2,49) has no subgroup with order 560 or 280, a contradiction.
Suppose T = PSL(2,224). Then Out(T) = Z24 by [12, P. 135], it follows A = PSL(2,224).Zr with r | 24, and |A| = 224-32-5-7-13-17-97-241-257-673r. Hence m = 343^ 17^97^241 ^257^673, 5-13-17-97-241-257-673 or 3^1347^24L257^673. For the first case, |Aa | = 223 3 5 7r, which is impossible by Lemma 2.6. For the second case, |Aa| = 223^32 7r, by Lemma 2.6, the only possibility is r = 2 and Aa = Z20 : (SL(2, 2) x SL(3, 2)); for the last case, we have |Aa| = 223-3-7r, by Lemma 2.6, the only possibility is r = 6 and Aa = Z2° : (SL(2,2)xSL(3, 2)). However, by Lemma 2.3, both PSL(2,224).Z2 and PSL(2, 224).Ze have no subgroup isomorphic to Z20 : (SL(2, 2) x SL(3,2)), which is a contradiction.
Suppose T = 3D4(2). Since Out(T) = Z3 (see [4]), A = 3D4(2) or 3D4(2).Z3, and so |A| = 212-34-72-13 or 212^7243 respectively, implying m = 3-7-13. Now, |Aa| = 2m = 211^33^7 or 2n-34-7, which is impossible by Lemma 2.6.
Arguing similarly as above, one may prove that no graph r exists for all other candidates for T in Table 4 (the results have been checked by Magma [1]).	□
We finally consider the case where A is insoluble and R = 1 by the following lemma.
Lemma 5.3. Suppose that A is insoluble, R =1 and t > 3. Then no graph r exists.
Proof. Let M be a minimal soluble normal subgroup of A. Then M = ZJ?, where r is a prime and d > 1. Since t > 3, M has at least 2 • 3 • 5 = 30 orbits on Vr, so, by Theorem 2.7, M is semi-regular on Vr (so d =1 and r G {2,p1,p2,... ,pt}), rM is a 7-valent A/M-arc-transitive graph of order 2m, and F is an arc-transitive regular Zr-cover of Fm .	r
If r = 2, then rM is arc-transitive of odd order m and odd valency 7, which is impossible.
Thus, r = pi with i G {1, 2,..., t}, and rM is a 7-valent A/M-arc-transitive graph of order . Recall that we assume by inductive hypothesis that Theorem 1.3 is true for all graphs which satisfy the assumptions of Theorem 1.3 and are of order less than n, so rM satisfies Theorem 1.3. Noting that A is insoluble and M is soluble, A/M is insoluble, so is Aut(rM). Then, checking the graphs in Theorem 1.3, we conclude that the soluble radical of Aut(rM) equals 1, and one of the following holds:
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(1)	rM == C330 and Aut(rM) = M22.Z2;
(2)	Aut(rM) = PSL(2,p) or PGL(2,p) withp > 13 aprimeandt > 3.
Since A/M acts arc-transitively on rM, we have 7|VrM | divides |A/M|.
For case (1), then |VrM| = 330, and 7 • 330 = 2310 divides |A/M|, since A/M < M22.Z2, by [4], we have A/M = M22 or M22.Z2. Let X be a normal subgroup of A such that X = M.M22 = Zr.M22. Since (r, |VrM|) = (r, 330) = 1, we have r = 2 and 3. Then, as |Mult(M22)| = 12 (see [4]), Lemma 2.5 implies X = Zr x M22 and X' = M22. Since |X'| = |M221 does not divide |Vr|, X' < A is not semi-regular on Vr, and X' has at most two orbits on Vr by Theorem 2.7. Because r is connected and 1 = X'a < Aa, 1 = (Xa)r(a) < Ai(a), it follows 7 | X |. Hence r divides ^ = 27 • 32 • 5 • 11, which is a contradiction as (r, |VrM |) = (r, 330) = 1.
We next consider case (2). Since A/M is insoluble and 7 | |A/M|, by Lemma 2.4, A/M = PSL(2,p) or PGL(2,p). Let B/M < A/M such that B/M ^ PSL(2,p). Since Mult(PSL(2,p)) = Z2 (see [12, P. 302]) and r > 3, Lemma 2.5 implies that B' = PSL(2,p) and B = M x B'. Since B,B' < A are insoluble, both B and B' have at most two orbits on Vr. In particular, m divides |B'|.
If r > 7, since |A| divides 225 • 34 • 52 • 7m, and |B| = |M x B'| = r|B'| divides |A|, we have r / |B'|, which is a contradiction to m dividing |B'|.
Assume r = 7. Since r is connected and 1 = B'a < Aa, we have 7 | |Bi |. Then, as |B':Bi | = m or 2m is divisible by 7, we further conclude that 72 divides |B'| = |B/M|. However, since |B/M:(B/M| = m or , which is not divisible by 7, we have 72 | |(B/M)a |, so 72 | |(a/m)s|, which is a contradiction by Lemma 2.6.
Assume finally r = 3 or 5. Since B/M = B' has at most two orbits on VrN, and B' has at most two orbits on Vr, we have |B/M:(B/M)s | = m or ^, and |B':Bi | = m or 2m. It follows that r | |(B/M)s| and so r | |(A/M)51. Also, as a/m acts arc-transitively on rM, 7 | |(A/M)51, hence 7r | |(A/M)51. Suppose r = 3. If (A/M)5 is soluble, then (A/M )5 is listed in part (1) of Lemma 2.6, and as 21 | |(A/M )s |, we have (A/M )s > F21; however, since A/M < PGL(2, p) and p = 7, by Lemma 2.4, PGL(2, p) has no soluble subgroup containing a subgroup isomorphic to F21, a contradiction. If (A/M)s is insoluble, noting that A/M < PGL(2,p), we have (A/M)s = PSL(2,p) or A5. For the first case, |VrM| = |A/M:(A/M)s| = 2, which is impossible. For the latter case, 7/ |(A/M)s|, also a contradiction. Suppose now r = 5. Then 35 | |(A/M)s|, and Lemma 2.6 implies that (A/M)s is insoluble, so (A/M)s = PSL(2,p) or A5 as A/M < PGL(2,p). Now, the same arguments as above draw a contradiction.	□
Theorem 1.3 now follows directly from Theorem 1.2 and Lemmas 5.1-5.3. References
[1]	W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: The user language, J. Symbolic Comput. 24 (1997), 235-265, doi:10.1006/jsco.1996.0125.
[2]	P. J. Cameron, G. R. Omidi and B. Tayfeh-Rezaie, 3-designs from PGL(2,q), Electron. J. Combin. 13 (2006), #R50, http://www.combinatorics.org/ojs/index.php/ eljc/article/view/v13i1r50.
[3]	Y. Cheng and J. Oxley, On weakly symmetric graphs of order twice a prime, J. Comb. Theory Ser. B 42 (1987), 196-211, doi:10.1016/0095-8956(87)90040-2.
64
Ars Math. Contemp. 15 (2018) 39-51
[4]	J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of Finite Groups, Oxford University Press, Eynsham, 1985, http://brauer.maths.qmul.ac. uk/Atlas/v3/.
[5]	L. E. Dickson, Linear Groups with an Exposition of the Galois Field Theory, Dover Publications, New York, 1958.
[6]	S.-Y. Ding, B. Ling, B.-G. Lou and J.-M. Pan, Arc-transitive pentavalent graphs of square-free order, Graphs Combin. 32 (2016), 2355-2366, doi:10.1007/s00373-016-1717-8.
[7]	J. D. Dixon and B. Mortimer, Permutation Groups, volume 163 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1996, doi:10.1007/978-1-4612-0731-3.
[8]	Y.-Q. Feng, C. H. Li and J.-X. Zhou, Symmetric cubic graphs with solvable automorphism groups, Eur. J. Combin. 45 (2015), 1-11, doi:10.1016/j.ejc.2014.10.008.
[9]	Y.-Q. Feng and Y.-T. Li, One-regular graphs of square-free order of prime valency, Eur. J. Combin. 32 (2011), 265-275, doi:10.1016/j.ejc.2010.10.002.
[10]	M. Giudici, C. H. Li and C. E. Praeger, Analysing finite locally s-arc transitive graphs, Trans. Amer. Math. Soc. 356 (2004), 291-317, doi:10.1090/s0002-9947-03-03361-0.
[11]	C. D. Godsil, On the full automorphism group of a graph, Combinatorica 1 (1981), 243-256, doi:10.1007/bf02579330.
[12]	D. Gorenstein, Finite Simple Groups, University Series in Mathematics, Plenum Press, New York, 1982.
[13]	S.-T. Guo, Y.-T. Li and X.-H. Hua, (G, s)-transitive graphs of valency 7, Algebra Colloq. 23 (2016), 493-500, doi:10.1142/s100538671600047x.
[14]	B. Huppert, Endliche Gruppen I, volume 134 of Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Berlin, 1967, doi:10.1007/978-3-642-64981-3.
[15]	B. Huppert and W. Lempken, Simple groups of order divisible by at most four primes, Proc. of the F. Skorina Gomel State Univ. 10 (2000), 64-75, http://www.iem.uni-due.de/ ~lempken/pi4.ps.
[16]	C. H. Li, Z. P. Lu and G. X. Wang, Vertex-transitive cubic graphs of square-free order, J. Graph Theory 75 (2014), 1-19, doi:10.1002/jgt.21715.
[17]	C. H. Li, Z. P. Lu and G. X. Wang, The vertex-transitive and edge-transitive tetravalent graphs of square-free order, J. Algebr. Comb. 42 (2015), 25-50, doi:10.1007/s10801-014-0572-z.
[18]	C. H. Li and J.-M. Pan, Finite 2-arc-transitive abelian Cayley graphs, Eur. J. Combin. 29 (2008), 148-158, doi:10.1016/j.ejc.2006.12.001.
[19]	B. Ling and S. Y. Ding, On 7-valent symmetric graphs of order 2pq, preprint.
[20]	A. Malnic, D. Marusic and P. Potocnik, On cubic graphs admitting an edge-transitive solvable group, J. Algebr. Comb. 20 (2004), 99-113, doi:10.1023/b:jaco.0000047284.73950.bc.
[21]	J.-M. Pan, Z.-H. Huang and S.-Y. Ding, Arc-transitive cyclic covers of graphs with order twice a prime, Discrete Math. 340 (2017), 811-816, doi:10.1016/j.disc.2016.11.018.
[22]	J.-M. Pan, Z.-H. Huang and Z. Liu, Arc-transitive regular cyclic covers of the complete bipartite graph Kp,p, J. Algebr. Comb. 42 (2015), 619-633, doi:10.1007/s10801-015-0594-1.
[23]	J.-M. Pan, Y. Liu, Z.-H. Huang and C.-L. Liu, Tetravalent edge-transitive graphs of order p2q, Sci. China Math. 57 (2014), 293-302, doi:10.1007/s11425-013-4708-8.
[24]	C. E. Praeger, An O'Nan-Scott theorem for finite quasiprimitive permutation groups and an application to 2-arc transitive graphs, J. LondonMath. Soc. 47 (1993), 227-239, doi:10.1112/ jlms/s2-47.2.227.
J. Pan et al.: On prime-valent symmetric graphs of square-free order
65
[25]	J. Schur, Untersuchungen über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen, J. Reine Angew. Math. 132 (1907), 85-137, doi:10.1515/crll.1907.132.85.
[26]	M. Suzuki, Group Theory II, volume 248 of Grundlehren der mathematischen Wissenschaften, Springer-Verlag, New York, 1986, doi:10.1007/978-3-642-86885-6.
[27]	R. Weiss, The nonexistence of 8-transitive graphs, Combinatorica 1 (1981), 309-311, doi: 10.1007/bf02579337.