Vertex-primitive groups and graphs of order twice the product of two distinct odd primes

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1 J. Group Theory 3 (000), 47±69 Journal of Group Theory ( de Gruyter 000 Vertex-primitive groups and graphs of order twice the product of two distinct odd primes Greg Gamble and Cheryl E. Praeger (Communicated by A. A. Ivanov) Abstract. A non-cayley number is an integer n for which there exists a vertex-transitive graph on n vertices which is not a Cayley graph. In this paper, we complete the determination of the non- Cayley numbers of the form pq, where p; q are distinct odd primes. Earlier work of Miller and the second author had dealt with all such numbers corresponding to vertex-transitive graphs admitting an imprimitive subgroup of automorphisms. This paper deals with the primitive case. First the primitive permutation groups of degree pq are classi ed. This depends on the nite simple group classi cation. Then each of these groups G is examined to determine whether there are any non-cayley graphs which admit G as a vertex-primitive subgroup of automorphisms, and admit no imprimitive subgroups. The outcome is that pq is a non-cayley number, where <q< p and q and p are primes, if and only if one of p11 mod 4, or q11 mod 4, orp11 mod q, orp ˆ 4q 1, or p; q ˆ 11; 7 or 19; 7 holds. 1 Introduction In 1983, MarusÏicÏ [18] asked for a determination of the set NC of natural numbers n for which there exists a vertex-transitive graph of order n, that is, on n vertices, which is not a Cayley graph. The elements of NC are called non-cayley numbers. Since the vertex-disjoint union of k copies of a non-cayley, vertex-transitive graph of order n is a non-cayley, vertex-transitive graph of order kn, it follows that any multiple of a non-cayley number is also a non-cayley number. Thus non-cayley numbers with few prime divisors are of particular interest. The question of membership of NC has been settled for all natural numbers which are not square-free [, 3], or are the product of two distinct primes [0]. The results of this paper contribute to the classi- cation, completed in [9], of the non-cayley numbers which are products of at most three primes. The problem of membership of NC for numbers of the form pq, where p and q are distinct odd primes, was studied in depth by Miller and the second author in [4], and the problem was solved except for the case of graphs for which every vertex-transitive subgroup of automorphisms is vertex-primitive. The purpose of this paper is to complete the determination of non-cayley numbers of the form pq. The strategy we use for this is to determine, rst, all primitive permutation

2 48 G. Gamble and C. E. Praeger groups of degree pq. Then each of these groups G is examined to determine whether there are any graphs with the above property which admit G as a vertex-primitive subgroup of automorphisms, and admit no imprimitive subgroups of automorphisms. Thus our results are of two types. First we have Theorem 1, the classi cation of primitive permutation groups of degree pq. This result depends on the nite simple group classi cation. Remark 1 (on notation). In all cases G will represent a primitive permutation group with socle T acting on a set W. Thus in particular T is transitive on W. Let a A W. Then the action of G on W is permutationally isomorphic to the action of G by right multiplication on the set G : G a Š of right cosets of the stabilizer G a in G. In our proofs it will sometimes be convenient to identify W with G : G a Š. Moreover, since also T is transitive on W, when considering the action of T on W we may similarly identify W with T : T a Š. In the tables summarizing the examples we will often specify T and W ˆ T : T a Š. We will generally refer to candidates or possibilities that satisfy our conditions as examples. Often a possibility is discounted because the number of prime divisors of the degree (counted according to multiplicity) of its associated action is less than, or more than three; for brevity, we will say, in such a case, that the degree has insu½cient, or excess divisors, respectively. The letter r always denotes a prime power (usually of r 0, but sometimes we require only that r 0 divides r), and s always denotes a prime. For r a power of a prime r 0, the r-part of an integer N, written jnj r, is the highest power of r 0 that divides N (so that jnj r ˆjNj r0 ). A group G is said to be almost simple if T W G W Aut T for some non-abelian simple group T. Often we will refer to Aschbacher classes of subgroups of a classical simple group T; these are the classes C 1 ;...; C 8 de ned in [1]. (Note that Kleidman and Liebeck choose to include the Aschbacher class C 0 1 [, Section 13] in their de nition of C 1;see [1, pp. 4, 58].) Also de ned in [1] is the set S (which we nd convenient to label C 9 ), a collection of quasi-simple subgroups of T. (A group is called quasi-simple if it is perfect and if G=Z G is a non-abelian simple group.) Notation for the classical simple groups follows [1]; we explain this and additional notation we use, including the term type, after Proposition 1 in Section. Finally, K m denotes the complete graph on m vertices. Theorem 1. Let p and q be distinct odd primes, where q < p, and suppose that G is a primitive permutation group on a set W of pq points. Then G is an almost simple group with a non-abelian simple socle T such that one of the following holds: (i) G is a not a classical group and T, p, q are given in one of the lines of Table 1; or (ii) G is a classical group and T a is an Aschbacher class C 1 subgroup, (i.e. apart from line 1 of Table, G a is the stabilizer of a totally singular subspace, or a nonsingular subspace) and T, p, q are given in one of the lines of Table ; or (iii) T ˆ PSL p, T a is not an Aschbacher class C 1 subgroup, and p, q are given in one of the lines of Table 3.

3 Vertex-primitive groups and graphs of certain orders 49 Table 1. Socles T of non-classical almost simple groups with primitive actions on W ˆ T : T a Š of degree pq, where < q < p and p; q are primes. T p q T a or action Comments A pq p q natural p 1 A p p W ˆ S, jsj ˆp px 13 4 p 1 A p 1 p W ˆ S, jsj ˆp 1 p X 11 4 A W ˆ S, jsj ˆ13 3 M 11 S 5 M M 10 : ( copies) M 11 < M 1 < A 1 on pairs M A 8 J PSL 11 Table. Socles T of classical almost simple groups with primitive actions on W ˆ T : T a Š of degree pq, where < q < p, p and q are primes, and T a is an Aschbacher class C 1 subgroup of T. T p q T a type Comments PSL P 1; G contains a graph automorphism PSL 5 r PSL 4 r PSL r PSU 3 r r 5 1 r 1 r 3 1 r 1 p r 3 1 r 1 PW m r r m 1 r 1 PW m r r m 1 r 1 r 1 r 1 p r 1 r m 1 1 r m 1 1 r 1 P r 1 1 mod 10, r prime, r X 9 P r ˆ 3; or r ˆ 5; or r 1 11; 9 mod 30 and r prime, r X 59 P 1 r 1 1 mod 4, r prime, r X 9 P 1 r ˆ s l, l X 0, s prime, 1 mod 1 if l ˆ 0 s 1 1; 49 mod 60 if l > 0 P 1 m ˆ l 1, l X 1, r and m prime, r X 3 P 1 m ˆ l, l X 1, r and m 1 prime, r X 3

4 50 G. Gamble and C. E. Praeger Table 3. Primes p; q for which T ˆ PSL p is the socle of a primitive group on W ˆ T : T a Š of degree pq, where < q < p; p and q are primes, and T a < T is of Aschbacher class C ±C 9. p q T a class Comments p p p 1 4 p 1 4 D p 1 C T < A p 1 on pairs, p X 11 D p 1 C 3 T < A p on pairs, p X S 4 C A 5 C 9 The fact that G is almost simple follows from the O'Nan±Scott Theorem (see, for example, [14]), using no more than the arithmetic properties of jwj ˆpq. Ifq < p, then the classi cation follows from work of Liebeck and Saxl [17]. Also, in the case where G is not a nite classical group, the classi cation was completed by the rst author in his PhD thesis [8], which addressed a much broader research question, namely the question of determining all nite primitive permutation groups which contain an element of prime order p having at most p cycles of length p. (This latter classi cation was completed in [8] except for the groups of a½ne type and the classical almost simple groups.) These `non-classical' examples will be derived from [8] in Section, and after this we shall proceed to analyse the case of the classical almost simple groups. Now every vertex-transitive graph G with G a vertex-transitive subgroup of automorphisms is a generalized orbital graph for the permutation group induced by G on the vertex set W. A generalized orbital graph for a transitive permutation group G on a set W is de ned as follows. The group G induces a natural action on W W by u; v g :ˆ u g ; v g for all u; v A W; g A G. A generalized orbital S is a non-empty G- invariant subset of W W, and the generalized orbital digraph corresponding to S is de ned as the digraph with vertex set W and arcs the ordered pairs in S. This digraph admits G as a vertex-transitive group of automorphisms. We obtain a simple undirected graph by choosing S to be self-paired (in the sense that u; v A S if and only if v; u A S); we avoid the diagonal (that is v; v B S for all v), and we identify each pair of arcs u; v and v; u with the undirected edge fu; vg. Thus the vertex-transitive graphs of order pq such that every vertex-transitive subgroup of automorphisms is vertex-primitive all arise as generalized orbital graphs for primitive permutation groups of degree pq, that is the groups involved must be among those classi ed in Theorem 1. The possible values of p and q for such groups are classi ed in Theorem below. Its proof will be given in Section 3.

5 Vertex-primitive groups and graphs of certain orders 51 Table 4. Primes p; q for which there exists a graph G on pq vertices such that Aut G and all vertextransitive subgroups of Aut G are vertexprimitive. p q Action of Aut G p p 1 4 S p 1 on pairs S 13 on triples 3 11 M 3 on cosets of A J 1 on cosets of PSL PSL p on cosets of S PSL 41 on cosets of A PSL 3 5 on cosets of P 1; r 3 1 r 1 r 5 1 r 1 r m 1 r 1 r m 1 r 1 r 1 r m 1 1 r m 1 1 r 1 PSL 4 r on cosets of P PSL 5 r on cosets of P PW m r on cosets of P 1, m an odd prime PW m r on cosets of P 1, m 1 an odd prime Theorem. Let p and q be distinct odd primes with q < p. Suppose that G is a graph on pq vertices such that Aut G is vertex-primitive, and moreover every vertex-transitive subgroup of Aut G is vertex-primitive. Then p, q are as in Table 4. Conversely there are examples of such graphs corresponding to each line of Table 4. Theorem provides a characterization of integers pq such that a vertex-transitive graph G of order pq exists for which all vertex-transitive subgroups of automorphisms are vertex-primitive. We remark that such a characterization is not strictly necessary for the determination of the non-cayley numbers of the form pq, since to supplement the results of [4] we only require knowledge of such numbers for which there exists a vertex-primitive non-cayley graph of order pq. However, Theorem is of some interest in its own right. To our knowledge, the rst construction of a vertex-transitive graph such that all vertex-transitive subgroups of automorphisms are vertex-primitive was given in [7]. Theorem gives several in nite families of such graphs.

6 5 G. Gamble and C. E. Praeger Finally we draw together the results of this paper and those of [4] to complete the determination of necessary and su½cient conditions for pq to belong to NC. Theorem 3. Let p and q be odd primes with q < p. (a) Some proper divisor of pq lies in NC (and hence also pq A NC) if and only if one of the following holds: (i) p 1 1 mod 4 or q 1 1 mod 4 ; (ii) p 1 1 mod q ; (iii) p ˆ 11, q ˆ 7. (b) pq A NC but no proper divisor of pq lies in NC if and only if p1q13 mod 4, and one of the following holds: (i) p 1 1 mod q and p 1 mod q ; (ii) q ˆ 1 4 p 1 ; (iii) p ˆ 19, q ˆ 7. Primitive groups of degree pq Let G; T; W be as in Theorem 1, and suppose rst that T is not a classical simple group. Proposition 1. If T is not a classical simple group, then T; p; q are as in one of the lines of Table 1. Proof. Firstly, we have T ˆ A pq under the natural action. Otherwise, the examples in Table 1 follow from the results summarized in [8, Table 3.3] and detailed in [8, Chapter 6]. Note that the entries of [8, Table 3.3] are listed in order of log p m (where, in our case, m ˆ q); so, only the possibilities up to the line with entry 1.48 in the column headed log p m (since, in our case p is at least 5, and log p q < log p p ˆ 1 log p ) and the families of possibilities with entry `>1' in the log p m column, need to be considered. Most of the non-classical possibilities listed in [8, Table 3.3] are easily discounted on trivial grounds, for example, because the degree of the action is odd or not square-free or has excess divisors. We only give the arguments for those candidates that are excluded for somewhat non-trivial reasons. Lines and 3 of Table 1, A p and A p 1, are established as follows: [8, Table 3.3] lists T ˆ A c acting of c degree on unordered pairs from a set of size c. Thus pjc or pjc 1, whence p A fc; c 1g, since jwj ˆpq with p the larger odd prime divisor. In a similar manner, in line 4 we see that the only example with T ˆ A c acting on triples occurs for c ˆ 13. Lines 5±8 of Table 1 are straightforward to establish from [8, Table 3.3].

7 Vertex-primitive groups and graphs of certain orders 53 Now consider G r, acting of degree r 6 1 = r 1. Since the degree is even, r must be odd. This leads to jwj ˆ r r 1 r 1 r r 1 which has excess divisors and so does not give an example in Table 1. Similarly, the G r action of degree r 6 1 r 1 = r 1 is discounted. We assume for the rest of this section that T is a classical simple group of dimension n X over the eld F r (or over F r in the unitary case), so that T is PSL n r, PSp n r (with n even, n X 4), PW e n r (with either e ˆG, n X 8, and n even, or e ˆ, n X 7 and nr odd), or PSU n r (with n X 3). When we wish to suppress the parameters n; r of a classical simple group we will use the term type; and we use a (roman) T for variable references to a type. For example, `T ˆ T n r where T is of type PSL' means `T ˆ PSL n r '. Note that, when r is odd, SO n r contains no non-trivial scalars and hence PW n r G W n r, and so we abbreviate the type PW to W or just W. Let ^T denote the corresponding subgroup of GL n r (except for the unitary case), that is, for T of types PSL, PSp or PW e, ^T ˆ SL n r, Sp n r or W e n r respectively, and in the unitary case, ^T ˆ SU n r < GL n r. Let ^T a be the preimage in ^T of the stabilizer T a, so that j ^T : ^T a jˆjt : T a jˆpq: Whenever G corresponds to a subgroup of PGL n r, ^G denotes a preimage of G containing ^T. Thus corresponding to the groups G; T; T a we have the preimages ^G; ^T; ^T a, where ^T a should be read as `T a -hat' (rather than a point-stabilizer of ^T). We shall usually analyse the various possibilities according to the nature of the ^T a - action on the underlying vector space V ˆ V n r. In most cases G corresponds to a subgroup of PGL n r. However, this is not always the case: the exceptional cases occur when (i) T ˆ PSL n r with n X 3ifG contains an element which interchanges k-spaces and n k -spaces, (ii) T ˆ PSp 4 r if G contains a graph automorphism, and (iii) T ˆ PW 8 r if G contains a triality automorphism. Case (i) has been taken into account in Aschbacher's work [] and will be dealt with in our analysis of the general case. To a lesser extent this is true for the groups PSp 4 r, and not true at all for the groups PW 8 r in case (iii). So we deal with cases (ii) and (iii) now, before proceeding further. Note that, in case (ii), we may assume that r >, since PSp 4 0 G A 6, and this case has already been treated. In our analysis we shall refer to primitive prime divisors of numbers of the form r i 1; these are primes which divide r i 1 but which do not divide r j 1 for any j satisfying 1 W j < i. It was proved by Zsigmondy [3] in 189 that, if r and i are

8 54 G. Gamble and C. E. Praeger integers with r X, i X 3 and r; i 0 ; 6, then r i 1 has a primitive prime divisor. Also r 1 has a primitive prime divisor unless r ˆ l 1 for some l. Proposition. If T ˆ PSp 4 r with r X 3, then G contains no graph automorphisms. Proof. Let r 0 be the prime dividing r. Suppose that G contains a graph automorphism. Then the possibilities for ^T a are given in [15, p. 96] (see also [, Section 14]). If ^T a is a parabolic subgroup then j ^T : ^T a jˆ r 1 r 1 =gcd ; r 1 which is not of the form pq. On the other hand, if ^T a is of type O G r wr S, O r :, Sp 4 r 1 (with r ˆ r1 c with c prime), or Sz r (with r an odd power of ), then j ^T : ^T a j is divisible by r0. The groups G of type O 8 r containing a triality automorphism also need separate attention. We treat these below using the classi cation by Kleidman [10] of the maximal subgroups of PW 8 r. Proposition 3. If T ˆ PW 8 r with r X, then G contains no triality automorphisms. Proof. Suppose that G 0 ˆ T ˆ PW 8 r, sothatg 0 W G W Aut G 0. Suppose also that M is a maximal subgroup of G. Then the Results Matrix [10, Table I] lists all the possibilities for M 0 ˆ G 0 V M. Let r i denote a primitive prime divisor of r i 1. Note that r 4 exists and also r 6 exists except for r ˆ. If r ˆ then, for each of the possibilities of M, jg : Mj is either odd or divisible by 4. So we may assume that r >. For lines 1±8 of the Results Matrix, jg : Mj is divisible by r 6 r 4. For lines 9±14, M 0 W W 7 r and so jg : Mj is divisible by r 3 r 4 1 =gcd 4; r 4 1 ; and similarly, for all remaining lines of the Results Matrix, jg : Mj is divisible by r. Thus, in all cases jg : Mj is not square-free. We now proceed to the general analysis. The case in which ^T a is reducible on V gives rise to a number of examples and we consider this case rst. Note that, for 1 W k < n, the number of k-dimensional subspaces of V is sub n; k r :ˆ Yk iˆ1 r n 1 i 1 r i : 1 Proposition 4. If ^T a is reducible on V then we have the examples in Table. Proof. The cases that we must consider are summarized in [1, Table 4.1A]. Suppose rst that ^T ˆ SL n r, and that G contains a graph automorphism, so that n X 3. Then ^T a is the stabilizer of a pair of subspaces W; U of V such that dim W ˆ k and dim U ˆ n k with 1 W k W 1 n, and either W H U, k 0 1 n (case L: type P k; n k of

9 Vertex-primitive groups and graphs of certain orders 55 [1, Table 4.1A]) or V ˆ W l U (case L: type GL k r l GL n k r of [1, Table 4.1A]). In the former case, j ^T : ^T a jˆsub n; n k r sub n k; k r ; while in the latter case, j ^T : ^T a jˆsub n; k r :r k n k. Since j ^T : ^T a j is square-free, we must be in the former case, and since j ^T : ^T a j 1 mod 4, r is odd and k ˆ 1 (otherwise jwj has excess divisors or the smallest prime divisor is larger than ). Thus j ^T : ^T jˆrn 1 a r 1 rn 1 1 r 1 ˆ pq: One of the factors r j 1 = r 1 is an odd prime, and for this factor the exponent j must be an odd prime. The other factor is twice an odd prime, and is of the form r j 1 r j 1 = r 1 for some integer j; we must have j ˆ 1 and hence n ˆ 3, p ˆ r 3 1 = r 1 and q ˆ r 1 =. Moreover, r 0 3 (otherwise q ˆ ) and r 1 mod 3 (otherwise 3 is a proper divisor of p). Hence r 1 1 mod 3 and so 3jq. Thus q ˆ 3, r ˆ 5 and p ˆ 31. This is line 1 of Table. Thus we may now assume that G corresponds to a subgroup of PGL n r, and T a is a subgroup in C 1 nc1 0 (see [1, pp. 4, 58]). Suppose rst that ^T a is the stabilizer of a direct sum decomposition W L W L, where W is a non-singular k-dimensional subspace (case U: type GU k r L GU n k r, case S: type Sp k r L Sp n k r, case O e : type O e 1 k r L Oe n k r where e ˆ e 1e, or case O G : type Sp n r with k ˆ 1, of [1, Table 4.1A]). Then, in each case, we nd (as with case L: type P k; n k above) that j ^T : ^T a j is divisible by r and hence is not squarefree. All remaining examples in Aschbacher class C 1 are of type P k. We consider these for the various possibilities for T. Case 1. T ˆ PSL n r. Here T a is the stabilizer in T of a k-subspace of degree jt : T a jˆsub n; k r. Now k W and r is odd, since otherwise jt : T a j has excess divisors or is odd. Suppose rst that k ˆ 1. Then jt : T a jˆr n 1 r n 1 1 n mod : So n is even, say n ˆ m. Since jt : T a j 1 mod 4, we have r 1 1 mod 4 and m odd. But then jt : T a jˆ rm 1 r 1 r 1 rm 1 r 1 has excess divisors unless m ˆ 1, whence n ˆ, pq ˆ r 1 as in line 4 of Table. So assume that k 0 1; then we have r odd, k ˆ and degree r n 1 r n 1 1 r 1 r : 1

10 56 G. Gamble and C. E. Praeger Recall that r n 1 = r 1 1 n mod. Subcase 1(a). n is odd. Here p ˆ r n 1 = r 1 and q ˆ 1 rn 1 1 = r 1 : Since q is an integer, n 1 1 mod 4, whence n ˆ 4l 1, say, and so q ˆ rl 1 r l 1 r : 1 Since q is prime, l ˆ 1, i.e. n ˆ 5, q ˆ 1 r 1, and primality of p forces r to be prime and r 0 3; 5. If r 1G mod 5 then 5 is a proper divisor of q. Ifr 1 1 mod 5 then 5 is a proper divisor of p. Thus r 1 1 mod 10 and hence we have line of Table. Subcase 1(b). n is even. Here p ˆ r n 1 1 = r 1 and q ˆ 1 rn 1 = r 1 ; and we nd that n ˆ 4, q ˆ 1 r 1, p ˆ r 3 1 = r 1 and r is prime. Further, if r 0 3 and r 1G mod 5 then 5 is a proper divisor of q. Also, if r 1 1 mod 3, then 3 is a proper divisor of p. Hence either r; p; q ˆ 3; 13; 5 or 5; 31; 13, orr 1 11; 9 mod 30, and so we have line 3 of Table. Case. T ˆ PSU n r. Here our actions have degree Q k iˆ1 rn 1 i 1 n 1 i Q k jˆ1 rj 1 where n X 3 and 1 W k W b 1 nc and we nd that n ˆ 3 and p; q are as listed in line 5 of Table. Case 3. T ˆ PSp n r or T ˆ W n 1 r, n even, n X 4. Here our actions have degree Q k iˆ1 rn 1 i 1 Q k jˆ1 r j 1 where 1 W k W 1 n, and, with a similar argument to that of Case 1, we nd that k ˆ, in which case n > 4 gives a degree with excess divisors and n ˆ 4 gives a degree that is divisible by 4. So there are no examples in this case.

11 Vertex-primitive groups and graphs of certain orders 57 Case 4. T ˆ PW n r, n ˆ m X 8. Firstly, if k ˆ m then our actions have degree Y k 1 iˆ1 r m i 1 r i 1 which has excess divisors, since k 1 X 3. Thus k < 1 n ˆ m and our actions have degree r m 1 r m k 1 Yk iˆ1 r m i 1 r i 1 whence, in order to avoid excess divisors, we must have k ˆ 1. Thus jt : T a jˆ rm 1 r m 1 1 rm 1 ˆ rm 1 r 1 r 1 rm 1 1 ; from which we obtain line 6 of Table. Case 5. T ˆ PW n r, n ˆ m X 8, T a a C 1 type P k subgroup. Here our actions have degree r m 1 r m k 1 Yk iˆ1 r m i 1 r i 1 and a very similar line of reasoning to Case 4 yields line 7 of Table. Having dealt with reducible (i.e. the Aschbacher C 1 ) subgroups of T, we now deal with the irreducible non-quasi-simple (i.e. the Aschbacher C ;...; C 8 ) subgroups of T and the quasi-simple (which we label C 9 ) subgroups of T. See [1] for a fuller description of the classes C 1 ;...; C 8, and the C 9 (ˆS, in [1]) subgroups. Propositions 5 and 6 reduce the remaining problem to the case when n ˆ, which is dealt with in Proposition 7. Proposition 5. If T a is in one of the Aschbacher classes C ±C 8, then T ˆ PSL r, for some prime-power r. Proof. We show that there are no example pairs T; T a where T a is a C ±C 8 subgroup of T for n X 3. The information we need is in Tables 3.5.A±F of [1] and the Theorems 4:i: j speci ed by columns I and II of these tables. Our main strategy is to show that jt : T a j is not square-free, by showing that r divides jt : T a j (our usual argument for classes C ; C 3 ; C 4 ; C 7 ; C 8 ), or that r 0 divides jt : T aj where r ˆ r e 0 for some prime e (particularly for class C 5 )orthatjt : T a j is at least 4. For class C 6, we use [13, Theorem 4.] which bounds jt a j above by r n 4 ; by assuming jt a jˆ

12 58 G. Gamble and C. E. Praeger Table 5. jt : T a j r for T a a C subgroup of T, where n ˆ kl. T T a type jt : T a j r Conditions PSL n r GL k r wr S l rn n k = n X 3, k X 1, l X PSU n r GU k r wr S l PSU n r GU n= r : r n =4 jj r n even r n= n k = PSp n r Sp n r wr S l k even, l X GL n= r : r n=4 n= 1 r odd r n 1 = l k 1 = W n r O k r wr S l k; l X 3 O 1 r wr S n r n 1 = r an odd prime PW G n r Oe k r wr S l r n= n= 1 lk= k= 1 j l 1 j r e A f ; g; k even, l X r n= n= 1 l k 1 = r odd, e ˆ; k even, l X O 1 r wr S n r n= n= 1 r an odd prime O n= r r n= 1 n=4 1= r; n= odd PW n r GL n= r : r n=4 n= 1 jj r jtj= pq we deduce a contradiction except for a few easily-treated possibilities for which n is small. Let n X 3. We now consider each class in turn. The class C. The pairs T; T a that we need to consider are listed in Table 5; the values jt : T a j r are readily determined from the information contained in [1]. We consider just two of the cases listed in Table 5; for the remaining cases, it is straightforward to show that jt : T a j r is at least r. Suppose that T ˆ PSL n r with T a of type GL k r wr S l,ort ˆ PSU n r with T a of type GU k r wr S l. Then n ˆ kl X 3, k X 1, l X and

13 Vertex-primitive groups and graphs of certain orders 59 Table 6. jt : T a j r for T a a C 3 subgroup of T, where n ˆ ke and e is prime. T T a type jt : T a j r Conditions PSL n r GL k r e PSU n r GU k r e PSp n r PSp k r e r n n k = jej r n X 3 r n= n k = jej r GU n= r r n =4 r odd W n r O k r e r n 1 = e k 1 = jej r n X 9, k > 1 PW G n r GU n= r : O G k re r n=4 n= 1 j n=; j r r n= n k = j e; ej r O n= r r n= 1 n=4 1= r; n= odd jt : T a j r ˆ r n n k = = ; and since W r l 1 = r 1 we have jt : T a j r X r. Now consider T ˆ PSp n r with T a of type Sp k r wr S l. Then n ˆ kl X 4, k is even, l X and jt : T a j r ˆ r n= n k = =. For n X 6 we have jt : T a j r X r n= n=4 n= 1 > r : Thus we may assume that n ˆ 4 and r > (since PSp 4 0 G A 6 has already been treated). Then, for r odd, jt : T a j r ˆ r ; and for r even, divides r, whence jt : T a j X =j!j ˆ 3 : The class C 3. The pairs T; T a we need to consider and corresponding values of jt : T a j r are listed in Table 6. As with C, we take two cases to illustrate our arguments; it is straightforward to show that jt : T a j r X r for the remaining possibilities. Suppose that T ˆ PSL n r with T a of type GL k r e,ort ˆ PSU n r with T a of type GU k r e. Then which is always strictly greater than r. jt : T a j r ˆ r n n k = =jej r X r n n k = 1

14 60 G. Gamble and C. E. Praeger Table 7. jt : T a j r for T a a C 4 subgroup of T, where n ˆ kl. T T a type jt : T a j r Conditions PSL n r PSU n r GL k r n GL l r GU k r n GU l r r n= n 1 k= k 1 l= l 1 W k < l PSp n r Sp k r n O l r r n= k= l 1 = k even, l X 3, r odd Sp k r n O G l r r n= k= l= l= 1 k even, l X 3, r odd W n r O k r n O l r r n 1 = k 1 = l 1 = 3 W k < l PW n r Sp k r n Sp l r r n= n= 1 k= l= jj r W k < l O e1 k r n Oe l r r n= n= 1 k= k= 1 l= l= 1 e i A f ; g, k; l X 4 PW G n r OG k r n O l r r n= n= 1 k= k= 1 l 1 = k X 4; l X 3 Now suppose that T ˆ PSp n r with T a of type PSp k r e. Then n ˆ ke X 4, e is prime and For n X 6 we have jt : T a j r ˆ r n= n k = =jej r : jt : T a j r X r n= n=4 1 > r 3 : This leaves n ˆ 4 (whence k ˆ e ˆ ) and r > to check. For r odd, jt : T a j r ˆ r ; and for r even, divides r, whence jt : T a j X =jj ˆ 3. The class C 4. The cases to be considered here are listed in Table 7. Consider the case when T ˆ PSp n r with T a of type Sp k r n O e l r, where n ˆ kl X 6, k is even and l X 3, so that k W 1 3 n and l W 1 n. Thus, whether e ˆor e A f ; g, we have jt : T a j r X r n= n=3 = n= 1 = > r : The arguments for the other cases are similar. The class C 5. The cases to be considered are listed in Table 8. Where the table lists jt : T a j r0 it is straightforward to show that jt : T a j r0 X r0 ; in the remaining cases jt : T a j r is listed and for these cases it is easy to show that jt : T a j r X r. The class C 6. Here [13, Theorem 4.] bounds jt a j above by r n 4. Thus, for primes p; q such that < q < p, we rst assume that jt : T a jˆpq, or equivalently that

15 Vertex-primitive groups and graphs of certain orders 61 Table 8. jt : T a j r0 or jt : T a j r for T a a C 5 subgroup of T, where r ˆ r0 e, e prime. T T a type jt : T a j r0 or jt : T a j r Conditions PSL n r GL n r 0 PSU n r GU n r 0 r e 1 n n 1 = 0 PSU n r O n r r n 1 =4 r; n odd O G n r rn =4 r odd, n even Sp n r r n= n= 1 n even PSp n r PSp n r 0 r e 1 n= 0 W n r O n r 0 r e 1 n 1 = 0 PW G n r Oe n r 0 r e 1 n= n= 1 0 e A f ; g jt a jˆjtj= pq and so derive a lower bound for T a that contradicts [13, Theorem 4.] except for some small possibilities for n. Case 1. T ˆ PSL or PSU with T a of type s k Sp k s. Here n ˆ s k, s is prime and s 0 r 0, where r 0 is the prime divisor of r. For n X 5, assume for a contradiction that jt a jˆjtj= pq for distinct primes p; q. Then pq is certainly not larger than r n e r n e = r e, where e ˆ if T ˆ PSL or e ˆ if T ˆ PSU. Thus jt a jˆ jtj pq X 1 rn n 1 = r 1 r 4 1 X 1 rn r4 > r n 4 contradicting the upper bound of [13, Theorem 4.]. For n ˆ 4 we have T a A f 4 :A 6 ; 4 :Sp 4 g and r is odd, whence jt : T a j r ˆ r 6 =j6!j r ˆ r 6 =j3 :5j r X r 4 : For n ˆ 3 we have T a A f3 :Q 8 ; 3 :Sp 3 g, 3 does not divide r, and r 0. Hence jt : T a j r ˆ r 3 if r is odd, or jt : T a j X 6 = 3 ˆ 3 if r is even. Case. T ˆ PSp with T a of type 1 k :O k. Here 4 W n ˆ k and r is an odd prime. Firstly n < 8, by a similar argument to that of Case 1, and hence n ˆ 4: here k ˆ and hence whence jt : T a j r ˆ r =j5 :3j r X r. jt a jˆ 4 : ˆ 6 :5 :3;

16 6 G. Gamble and C. E. Praeger Table 9. jt : T a j r for T a a C 7 subgroup of T, where n ˆ k l. T T a type jt : T a j r Conditions PSL n r GL k r wr S l rn n 1 = lk k 1 = k X 3, l X PSU n r GU k r wr S l PSp n r Sp n r wr S l r n= l k= W n r O k r wr S l r n 1 = l k 1 = PW n r Sp k r wr S l r n= n= 1 l k= k X, l X 3 k X 3, l X k even, l X O G k r wr S l r n= n= 1 lk= k= 1 k X 4, l X Case 3. T ˆ PW with T a of type 1 k :O k. Here 8 W n ˆ k and r is an odd prime. Firstly n < 10, by a similar argument to that of Case 1. This leaves n ˆ 8to check: here jt a j r ˆj3 :5:7j r W r and hence jt : T a j r X r n= n= 1 ˆ r 10. The class C 7. For each C 7 type, n ˆ k l with at least the restriction that l X, k X. In particular, we have n ˆ k l X kl and l W n=k W 1 n. The cases we need to consider are listed in Table 9. We consider two cases; for the remaining cases it is straightforward to show that jt : T a j r X r. Suppose that T ˆ PSL n r with T a of type GL k r wr S l, or that T ˆ PSU n r with T a of type GU k r wr S l. Then n ˆ k l X 9, k X 3, l X and jt : T a j r ˆ rn n 1 = lk k 1 = X r n n k = l 1 X r n =4 n=3 1 > r : Suppose that T ˆ PW n r with T a of type Sp k r wr S l. Then 8 W n ˆ k l even and l X, and with k jt : T a j r ˆ r n= n= 1 l k= = : If n > 8 then n is at least 16, whence jt : T a j r X r.ifn ˆ 8 then k ˆ, l ˆ 3, jt : T a j r ˆ r 4:3 3 =j3!j r X r 8. The class C 8. The cases to be considered are listed in Table 10. Suppose rst that T ˆ PSL n r. For T a of type U n r 0 where the table lists jt : T a j r0 it is straightforward to show that jt : T a j r0 X r0 3. In the remaining PSL n r cases jt : T a j r X r.

17 Vertex-primitive groups and graphs of certain orders 63 Table 10. jt : T a j r0 or jt : T a j r for T a a C 8 subgroup of T T T a type jt : T a j r0 or jt : T a j r Conditions PSL n r Sp n r r n= n= 1 n even O n r r n 1 =4 r; n odd O G n r rn =4 r odd, n even U n r 0 PSp n r O G n r r n= r n n 1 = 0 r ˆ r 0 jj r r even Now suppose that T ˆ PSp n r with T a of type O G n r. Then r is even and jt : T a j r ˆ r n= =jj r.ifnx6thenjt: T a j r X r n= 1 X r.ifnˆ4 then since r > and r is even we have jt : T a j X 1 n= ˆ 3. Thus we have shown that either jt : T a j is not square-free or, in the case of C 6 groups, jtj= pq is greater than the jt a j lower bound provided by [13, Theorem 4.], and hence there can be no examples for T a a C ±C 8 subgroup of T for n >. Proposition 6. If T a is a quasi-simple subgroup of T in the Aschbacher class C 9, then T ˆ PSL r, for some prime-power r. Proof. Assume that n >. By [13, Theorem], if T a is a quasi-simple subgroup of T ˆ T n r then jt a j < r 3n or, in some cases, Ta 0 may be A c where c is n 1orn. For TˆPSp, W, PW or PW, we must check that Ta 00A c where c is n 1 orn. Since jtj r X r e where e X n n =4 and j n!j r W r d where d W n 1 = r 1 W n 1; we nd that at least r divides jt : T a j, if n X 7. If n < 7 then n ˆ 4 or 6 and T ˆ PSp, jtj r ˆ r n =4 ; again, in these cases, jt : T a j > r (since T 0 Sp 4 0 ). Thus we may assume that Ta 0 0 A c. Since we require jt : T a jˆjtj=jt a j to be of the form pq, most possibilities may be dealt with by showing, for any prime divisors p; q of jtj, that jtj= pq X r 3n. Indeed, this inequality holds in the following cases: n X 6 for T ˆ PSL or PSU, n X 8 for T ˆ PSp, n X 9 for T ˆ W, orn X 10 for T ˆ PW or PW. The argument for the case T ˆ PSL is as follows (the arguments for the other cases are similar). Suppose that T ˆ PSL n r and n X 6. Then, considering upper bounds for prime divisors of jtj, we see that pq is at most r n 1 r n 1 = r 1, whence jtj pq X 1 rn n 1 = r 1 r n 3 1 r n 1 1 X r 3n r 7 8 r r5 > r 3n :

18 64 G. Gamble and C. E. Praeger When n ˆ 5 for T ˆ PSL or PSU, [13, Theorem 4.] gives a better bound (jt a j < r n 4 ), and an argument similar to that above shows that jtj= pq > r n 4. Thus the following possibilities remain: n A f3; 4g for T ˆ PSL or PSU, n A f4; 6g for T ˆ PSp, n ˆ 7forTˆ W, and n ˆ 8 for T ˆ PW or PW. These remaining possibilities are discounted by checking the possible C 9 (ˆS, in [11]) subgroups listed in Kleidman's tables [11]. In each case, r divides jt : T a j. Now we are left with the case when T ˆ PSL r, with T a ˆ G a V T where G a is a maximal irreducible subgroup of G. All such groups were classi ed by Dickson [6]. (Here we include the quasi-simple possibility also, namely, T a ˆ A 5.) Since we have already considered the groups T ˆ A n, we may assume here that r X 7 and r 0 9. Proposition 7. If T ˆ PSL r, where r X 7 and r 0 9, then r ˆ p and the examples are listed in Table 3. Proof. We have several cases to check. In each case where there are examples we will nd that r ˆ p. Case 1. T a is a dihedral subgroup of T of index 1 r r 1 (resp. 1 r r 1 ). Here we obtain the examples listed in line 1 (resp. line ) of Table 3. Case. T a is a linear group over a sub eld of F r, i.e. T a ˆ PSL r 1=k :c where c ˆ gcd k; ; r 1, r ˆ r0 e and kje. Here jt : T a jˆ r r 1 cr 1=k r =k 1 : Observe that r 1 1=k ˆ r e=k k 1 0 must be prime. Hence r 0 is prime and e=k ˆ k 1 ˆ 1, i.e. k ˆ ˆ e. Now r 0 cannot be. So jt : T a jˆ1 r 0 r0 1 with r 0 odd, whence r0 1 1 mod 4 and 1 r mod which makes jt : T aj odd, and hence there are no examples in this case. Case 3. T a ˆ A 4 or S 4. Here r is an odd prime and jt : T a jˆr r 1 = 4c, where c is 1 if r 1G3 mod 8 or c is if r 1G1 mod 8. Hence p ˆ r and q ˆ r 1 = 48c ˆ r 1 r 1 = 48c : Since r 1 and r 1 di er by, one has -part at least 8c and the other. In particular, r 1G1 mod 8, sothatc ˆ. Thus, for e ˆ G1, we have r e =16 A f1; 3g, from which we obtain line 3 (e ˆ 1) and line 4 (e ˆ 1) of Table 3. Case 4. T a ˆ A 5. Here r 1G1 mod 10, r is prime and jt : T a jˆr r 1 =10. Hence p ˆ r and q ˆ r 1 =40. Suppose rst that jr 1j X 8 and jr 1j ˆ. Then either r 1 1 mod 10, whence r 1 =40 A f1; 3g; orr 1 1 mod 10 whence one of 1 8 r 1, 1 10 r 1 is 3. Each possibility leads to a contradiction. Hence jr 1j X 8 and jr 1j ˆ. Similar checks here yield the one example listed as the nal line of Table 3.

19 Vertex-primitive groups and graphs of certain orders 65 3 Non-Cayley numbers pq arising from vertex-primitive graphs In this section we examine each of the primitive permutation groups G on a set W of size pq (where < q < p and p, q are primes) occurring in Theorem 1 to decide whether or not there exists a graph G with vertex set W such that (i) G W Aut G, and (ii) every subgroup L of Aut G such that L is transitive on W acts primitively on W. First we outline the strategy we will use, and then we give the details of our arguments, which will prove Theorem. Let G W Sym W be primitive on W of degree pq and let G be a graph with vertex set W admitting G as a subgroup of automorphisms (with its given action on W) which is therefore vertex-primitive. From our discussion towards the end of Section 1, we know that G is a generalized orbital graph for G relative to some self-paired generalized G-orbital in W W. IfG were the complete graph K pq then Aut G ˆ S pq would contain transitive imprimitive subgroups (for example, there is a transitive cyclic subgroup of order pq), and so (ii) above would not hold. Hence we may assume that G 0 K pq. In particular, G is not -transitive on W, and hence G is not one of the groups in Table 1, line 1 or in Table, lines 4 or 5. Our rst task is to identify A :ˆ Aut G. Since A contains G, A is primitive of degree pq and hence is almost simple and its socle and stabilizer are listed in one of the the lines of Tables 1, or 3. Often (but not always) we will nd that A ˆ G. Once we have determined A we will check that the socle of A is primitive on W (since otherwise (ii) would not hold), and then decide whether there exist any proper vertex-transitive subgroups L of A (not containing the socle of A), and if so whether such subgroups can act imprimitively on W. IfL is such a subgroup then we have a proper factorization A ˆ LA a of the almost simple group A, where neither L nor A a contain the socle of A, and at least A a is a maximal subgroup of A. IfM is maximal such that M contains L and M does not contain soc A, thena ˆ MA a and either M is a maximal subgroup of A and all possibilities for A; M; A a are listed in [15, Tables 1±6 and Theorem D], or else the possibilities for A; M; A a are given in [16]. In some cases we will not need to resort to these tables as the fact that pq divides jlj may be su½cient to identify all possibilities for L. In yet other cases, when A is su½ciently small, we are able to nd the information we need in [5]. Thus we are able to determine whether A has any transitive imprimitive subgroups. We now consider the possibilities for G, occurring in Tables 1, and 3, line by line, and nd all p; q (and representative groups/graphs) satisfying (i) and (ii). Let G W Aut G where G has vertex set W, and G is not a complete graph. Table 1, line 1. Since G is -transitive this gives no examples, as discussed above.

20 66 G. Gamble and C. E. Praeger Table 1, line. The group T ˆ A p has rank 3 with T a -orbits of lengths 1, p, p. Since G is not a complete graph, it follows (for example by checking Tables 1, and 3) that A ˆ Aut G ˆ S p. A Frobenius subgroup Z p :Z p 1 is transitive and imprimitive on W. Table 1, line 3. As in the previous case, T has rank 3, and since G 0 K p p 1 =,we have A ˆ Aut G ˆ S p 1. Let L < A, with L transitive on W. Then 1 p p 1 divides jlj, and L is transitive on pairs from the set S of size p 1 on which A acts naturally. It follows that L is transitive on S, and (since p divides jlj) in fact L is -transitive on S. Since p 1 ˆ 4q is not a prime power, L is therefore an almost simple -transitive group of degree p 1onS. (In particular, L is not a Frobenius group in its action on S, and hence L is not regular on W. Thus G is not a Cayley graph.) In fact (see, for example [4, p. 8]) L is PSL p, PGL p, M 11 or M 1 (with p ˆ 11), A p 1,orS p 1. All these groups are primitive on W and so we have line 1 of Table 4. Table 1, line 4. T ˆ A 13, jwj ˆ86 ˆ :11:13. Since G 0 K 86 it follows from Theorem 1 that A ˆ Aut G ˆ S 13. Let L W A, with L transitive on W. Then divides jlj and so L is 3-transitive of degree 13, and hence L contains A 13. Thus we have line of Table 4. Table 1, lines 5, 6. Here jwj ˆ66 and either T ˆ M 11 has rank 4 with subdegrees (i.e. T a -orbit lengths) 1, 15, 0, 30, or T ˆ M 1 has rank 3 with subdegrees 1, 0, 45 (see, for example, [6, pp. 35 and 38]). Moreover (by Theorem 1, essentially) either Aut G ˆ S 1 on pairs, or Aut G ˆ M 11 (and G has valency 15, 30, 35, or 50). In each case, Aut G W S 1 in its action on pairs, and the argument for Table 1 line 3 shows that there are examples as in line of Table 4. We note that the graphs arising when T ˆ M 1 are the same as those for Table 1 line 3, while additional graphs occur when T ˆ M 11. However, we do not of course obtain new values for p; q. Table 1, line 7. Here G ˆ T ˆ M 3 and we deduce from Theorem 1 that Aut G ˆ M 3. A transitive subgroup L of M 3 has order divisible by :11:3 and it follows from [5, p. 71] that L ˆ M 3. Thus we have an example: line 3 of Table 4. Table 1, line 8. Here G ˆ T ˆ J 1 and we deduce from Theorem 1 that Aut G ˆ J 1.A transitive subgroup L of J 1 has order divisible by.7.19 and it follows from [5, p. 36] that L ˆ J 1, giving line 4 of Table 4. Table, line 1. Here T ˆ PSL 3 5 and G ˆ Aut T ˆ T:; by Theorem 1, A ˆ Aut G ˆ T:. Let L < A, L transitive on W. By [5, p. 38] the only proper subgroup L with order divisible by :3:31 is L ˆ 31 : 6, the normalizer of a Singer cycle. We claim, however, that this group is not transitive on W. It is su½cient to show that an involution in L xes a point of W. From the fact that L ˆ 31 : 6 we know that L contains an involution g A AnT. From the character table [5, p. 38], g is in class B and C T g has order 10, and

21 Vertex-primitive groups and graphs of certain orders 67 (from the table of maximal subgroups in [5, p. 38]) C T g G S 5 is the stabilizer of a conic C in the projective plane PG 5. We note that W is the set of pairs U; W of subspaces of a 3-dimensional vector space over F 5 where dim U ˆ 1, dim W ˆ, and U H W. Thus W can be identi ed with the set of incident point-line pairs in PG 5. Now C is a set of 6 points, no two collinear. There are 15 secant lines to C (meeting C in two points), 6 tangent lines, and 10 external lines to C, and C T g leaves invariant each of these three subsets of lines. The element g interchanges points and lines of PG 5, and so maps C to a subset C g of 6 lines which is invariant under C T g. It follows that C g is the set of 6 tangent lines to C. Let a A C and let l be the tangent line to C on a. Then l ˆ b g for some b A C. Since C T g ˆS 5 acts - transitively on the 6 points of C, there is an involution h A C T g which interchanges a and b. Note that a hg ˆ b g ˆ l, and l hg ˆ a hg ˆ a, since hg ˆ h g ˆ 1 (since hg ˆ gh). Thus hg is an involution in AnT and hg xes the point a; l of W. By [5, p. 38] again, there is only one conjugacy class of involutions contained in AnT, the class B, and hence hg and g are conjugate in A. Thus g also xes a point of W. Thus we have proved that A satis es (ii), giving line 8 of Table 4. Table, lines ±7. We note rst that in lines 4 and 5, T is -transitive, so no examples arise here. Consider the other lines. Since G is not a complete graph, it follows from Theorem 1 and [15] that A ˆ Aut G has socle T. Suppose that L W A and L is transitive on W. Then pq divides jlj and we have A ˆ LA a with A a of type P for lines ±3 or P 1 for lines 6±7. By [15, Tables 1±6] and [14] there are no possibilities for L except subgroups containing T. Thus we have lines 9±1 of Table 4. Table 3. Here T ˆ PSL p. The values for p; q in line 1 already occur in line 1 of Table 4. In line, T a ˆ D p 1 and we have T ˆ T a P 1 (where P 1 is a parabolic subgroup of T). The subgroup P 1 is transitive and imprimitive on W, so we have no examples from this line. In the nal lines 3±5 the subgroup T a is S 4 or A 5, and no proper subgroup of A (ˆT or T:), not containing T, is transitive on W. This gives lines 5±7 of Table 4. This completes the proof of Theorem. 4 Proof of Theorem 3 It follows from [1, 7] that p A NC (respectively q A NC) if and only if p 1 1 mod 4 (respectively q 1 1 mod 4 ). Also, necessary and su½cient conditions for pq to lie in NC are given in [3, Theorem 1]. (We note that in parts (b), (c) and (d) of [3, Theorem 1] one of p 1, q 1 is divisible by 4 so that (a)(i) holds.) Thus part (a) is proved. Suppose then that pq A NC but that no proper divisor lies in NC. Then by part (a), p 1 q 1 3 mod 4. If there is a non-cayley vertex-transitive graph of order pq which has a transitive imprimitive subgroup of automorphisms then, by [4, Theorem ] we have p 1 1 mod q (and, of course, p 1 mod q by part (a)), or p; q ˆ 11; 3 which satis es q ˆ 1 4 p 1. On the other hand, if there is a non-cayley

22 68 G. Gamble and C. E. Praeger vertex-transitive graph of order pq such that every vertex-transitive subgroup of automorphisms is vertex-primitive, then by Theorem, p and q are as in one of the lines of Table 4. In line 1, we have q ˆ 14 p 1. Lines, 6, 7 do not occur, since mod 4. For lines 3, 5, 8, condition (b)(i) holds, while for line 4 condition (b)(iii) holds. This leaves the in nite families. We note that, if t is any odd integer, then 1 t mod 4. Thus in lines 9, 10, 11 we have q 1 1 mod 4, while in line 1, p 1 1 mod 4. Thus one of (b)(i)±(iii) holds. Conversely, if p; q are as in (b)(i) or in (b)(ii)±(iii), then pq A NC by [4, Theorem ] or Theorem respectively. Acknowledgment. The authors would like to thank the referees for their comments and suggestions. They also thank Tim Penttila for suggesting the geometrical argument used for Table, line 1, in the proof of Theorem. References [1] B. Alspach and R. J. Sutcli e. Vertex-transitive graphs of order p. In Second international conference on combinatorial mathematics (New York, 1978) (New York Acad. Sci., 1979), pp. 18±7. [] M. Aschbacher. On the maximal subgroups of nite classical groups. Invent. Math. 76 (1984), 469±514. [3] N. L. Biggs. Algebraic graph theory. Cambridge Tracts in Math. 67 (Cambridge University Press, 1974). [4] P. J. Cameron. Finite permutation groups and nite simple groups. Bull. London Math. Soc. 13 (1981), 1±. [5] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson. An atlas of - nite groups (Clarendon Press, 1985). [6] L. E. Dickson. Linear groups with an exposition of the Galois theory (Dover, 1958). [7] R. Frucht, J. E. Graver and M. E. Watkins. The groups of the generalized Petersen graphs. Proc. Cambridge Philos. Soc. 70 (1971), 11±18. [8] G. Gamble. Elements of prime order in primitive permutation groups. Ph.D. thesis. University of Western Australia (1996). [9] M. A. Iranmanesh and C. E. Praeger. On non-cayley vertex-transitive graphs of order a product of three primes. Preprint (1999). [10] P. B. Kleidman. The maximal subgroups of the nite 8-dimensional orthogonal groups. PW 8 q and of their automorphism groups J. Algebra 110 (1987), 173±4. [11] P. B. Kleidman. The subgroup structure of some nite simple groups. Ph.D. thesis. University of Cambridge (1987). [1] P. B. Kleidman and M. W. Liebeck. The subgroup structure of the nite classical groups. London Math. Soc. Lecture Note Series 19 (Cambridge University Press, 1990). [13] M. W. Liebeck. On the orders of maximal subgroups of the nite classical groups. Proc. London Math. Soc. (3) 50 (1985), 46±446. [14] M. W. Liebeck, C. E. Praeger and J. Saxl. On the O'Nan±Scott theorem for primitive permutation groups. J. Austral. Math. Soc. Ser. A 44 (1988), 389±396. [15] M. W. Liebeck, C. E. Praeger and J. Saxl. The maximal factorizations of the nite simple groups and their automorphism groups. Mem. Amer. Math. Soc. 43 (American Mathematical Society, 1990).

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