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A Family of Quasiprimitive 2-arc Transitive Graphs which Have Non-quasiprimitive Full Automorphism Groups
Europ. J. Combinatorics (1998) 19, 499–502 Article No. ej970194
A Family of Quasiprimitive 2-arc Transitive Graphs which Have Non-quasiprimitive Full Automorphism Groups C AI H ENG L I For each prime p ≡ 1 or −1(mod 24), we construct a (PSL(2, p), 2)-arc transitive graph 0 of valency 4 such that Aut 0 = PSL(2, p) × Z 2 . Thus there exists an infinite family of quasiprimitive 2-arc transitive graphs which have non-quasiprimitive full automorphism groups. c 1998 Academic Press
Let 0 = (V 0, E0) be a finite connected graph with vertex set V 0 and edge set E0. For a positive integer s, an s-arc of 0 is an (s + 1)-tuple (v0 , v1 , . . . , vs ) of vertices such that (vi−1 , vi+1 ) ∈ E0 for 1 ≤ i ≤ s and vi−1 6= vi+1 for 1 ≤ i ≤ s − 1. The graph 0 is said to be (G, s)-arc transitive if G ≤ Aut 0 and G is transitive on V 0 and on the set of s-arcs of 0. For an intransitive normal subgroup N of G, the quotient graph 0 N induced by N is defined as the graph with vertices the N -orbits on V 0 such that two N -orbits 11 and 12 are adjacent in 0 N if some vertex of 11 is adjacent in 0 to some vertex of 12 . It is shown in [9, Theorem 4.1] that if 0 is a non-bipartite (G, 2)-arc transitive graph then 0 N is a (G/N , 2)-arc transitive graph. Thus 0 is a cover of 0 N . If N is a maximal intransitive normal subgroup of G, then every non-trivial normal subgroup of G/N is transitive on V 0N . Let G be a transitive permutation group on . If there exists no non-trivial G-invariant partition of , then G is said to be primitive on ; if every non-trivial normal subgroup of G is transitive on then G is said to be quasiprimitive. Thus a primitive group is quasiprimitive, but the converse is not necessarily true, see [10]. Praeger [9] obtained a version of the O’Nan–Scott theorem for quasiprimitive groups, which divides quasiprimitive groups into eight classes, and then she proved that only four of them can act 2-arc transitively on a graph. This suggests a reasonable program to describe non-bipartite 2-arc transitive graphs by two steps: (i) describing graphs admitting a quasiprimitive 2-arc transitive group of automorphisms, (ii) characterizing 2-arc transitive covers of examples from (i). There has been some work related to this program, see for example [1, 3, 6, 7]. To determine isomorphism classes of quasiprimitive 2-arc transitive graphs, it is necessary to determine the full automorphism group of a quasiprimitive 2-arc transitive graph. However, this is a very difficult problem. In this note we investigate a question relevant to this problem. For a (G, 2)-arc transitive graph 0, G ≤ Aut 0 ≤ Sym(V 0), where Sym(V 0) is the symmetric group on V 0. If G is primitive then Aut 0 must be primitive. Thus a natural question arises (refer to Baddeley [1, Section 6] or Praeger [11, Section 7]): if G is quasiprimitive on V 0, is Aut 0 quasiprimitive? Baddeley [1, Section 6] constructed a single counterexample which is a graph with a group of twisted-wreath-action type (in terms of [9]) such that there exists a non-quasiprimitive group H satisfying G < H ≤ Aut0. In this paper we give the first infinite family of counterexamples such that Aut0 is not quasiprimitive. 0195-6698/98/040499 + 04
$25.00/0
c 1998 Academic Press
500
Cai Heng Li
For a group G and a core-free subgroup H of G, if there exists an element g ∈ G such that g 2 ∈ H and G = hH, gi, then we have a connected G-arc transitive graph 0 := 0(G, H, g) with V 0 = {H x | x ∈ G} and E0 = {{H x, H y} | x y −1 ∈ H g H }. Denote by [H : H ∩ H g ] the set of cosets of H ∩ H g in H . Then 0 is (G, 2)-arc transitive iff H acts 2-transitively on [H : H ∩ H g ] (see for example [3, Theorem 2.1]). The result of this paper is the following theorem. T HEOREM . Let p be a prime such that p ≡ 1 or −1 (mod 24). Let G = PSL(2, p). Then there exists a (G, 2)-arc transitive graph 0 of valency 4 such that G is quasiprimitive on V 0 while Aut 0 = G × Z2 and in particular Aut 0 is not quasiprimitive on V 0. P ROOF. Since p ≡ ε = ±1 (mod 24), p ≥ 23. Then by [12, p. 417], each maximal subgroup of G is isomorphic to one of the following: S4 , A5 , D p−1 , D p+1 , Z p ×| Z p−1 . 2
Note that |G| = p( p + 1)( p − 1)/2, and ( p − 1, p + 1) = 2. It follows that a Sylow 3subgroup G 3 of G is conjugate to a subgroup of D p−ε . Thus G 3 is cyclic and all subgroups of G of order 3 are conjugate. Let M be a subgroup of G isomorphic to S4 , and let H be a subgroup of M isomorphic to A4 . We shall construct a class of (G, 2)-arc transitive graphs of valency 4 with H the stabilizer of a point. (Recently, the author became aware that this class of graphs has been constructed in [6]. However, for the sake of completeness and in need for notation in the ensuing, we present a construction here.) By the discussion before the theorem it is sufficient to find an element g ∈ G such that hH, gi = G, g 2 ∈ H , |[H : H ∩ H g ]| = 4 and H is 2-transitive on [H : H ∩ H g ]. Now M and H may be written as M = (N ×| L)×| hbi and H = N ×| L, where N ∼ = Z22 , ∼ ∼ ∼ L = Z3 and hbi = Z2 such that hL , bi = S3 . Consider NG (L). By our observation above, NG (L) ∼ = D p−ε . Consequently, NG (L) can be written as NG (L) = (L 0 × hci)×| hbi for some c ∈ G, where L ≤ L 0 , L 0 is a Sylow 3-subgroup of G, and |L 0 |o(c) = ( p − ε)/2. Let g be the involution of hci. Then g centralizes both L 0 and b. Suppose that hH, gi < G. Then there exists a maximal subgroup M0 of G containing hH, gi. Since g ∈ / H , hH, gi > H ∼ = S4 or A5 . = A4 . It follows from the list (∗) that M0 ∼ However, since C M0 (L) ≥ ChH,gi (L) ≥ L × hgi ∼ = Z6 , we have that M0 6 ∼ = S4 , A5 , which is a contradiction. Therefore, hH, gi = G. 2 g Suppose that N0 := N ∩ N g 6= 1. Since g 2 = 1, N0 = N g ∩ N g = N0 . So g normalizes hL , b, N0 i. Since L is transitive on the non-identity elements of N by conjugation, hL , b, N0 i = M, and hence g normalizes M, which is a contradiction since hM, gi = G is simple. Therefore, N ∩ N g = 1 and so H ∩ H g = L. It follows that |[H : H ∩ H g ]| = 4 and H is 2-transitive on [H : H ∩ H g ]. Let 0 = 0(G, H, g). Since g 2 = 1, g 2 ∈ H . Thus 0 is a connected (G, 2)-arc transitive graph (by the discussion before the theorem) and G is quasiprimitive on V 0 (since G is simple). Set F := G × hσ i where hσ i ∼ = S4 and = Z2 . Let K = (N ×| L)×| hbσ i. Then K ∼ hK , gi = hH, bσ, gi = hG, bσ i = F. Since g centralizes both b and σ , g centralizes bσ , and further, since g centralizes L, g normalizes hL , bσ i. Since N ∩ N g = 1, K ∩ K g = hL , bσ i ∼ = S3 . It follows that |[K : K ∩ K g ]| = 4 and K is 2-transitive on [K : K ∩ K g ]. Thus 6 := 0(F, K , g) is a connected (F, 2)-arc transitive graph. Let v be the vertex of 6 corresponding to K . Since G < F and G v = Fv ∩ G = K ∩ G = N ×| L = H , it follows that 6 = 0. Since hσ i F, F is not quasiprimitive on V 0.
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Quasiprimitive 2-arc transitive graphs
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Finally, we prove that Aut 0 = F = G × hσ i so that Aut 0 is not quasiprimitive on V 0. Let A = Aut 0. Then F ≤ A and A is a transitive permutation group on V 0. Since 0 is of valency 4, every prime divisor is not greater than 4 so that Av is a {2, 3}-group (see [8, Lemma 2.1]). Suppose that A is quasiprimitive on V 0. By [9, Theorem 1.1], B := soc(A) = T1 × · · · × Tk where k ≥ 1 and T1 ∼ = ··· ∼ = Tk is simple. Since B A, we have G ∩ B G. Since A is quasiprimitive on V 0, B is transitive on V 0 and C A (B) ≤ B. Therefore, G ≤ B. Further, since T1 B, G∩T1 G, and so either G∩T1 = 1 or G∩T1 = G. Since Av is a {2, 3}-group, Bv is a {2, 3}-group. If G ∩ T1 = 1 then |hG, T1 i| = |G||T1 | dividing |A|, which is a contradiction since |A|/|G| = 2r 3s for some r, s ≥ 0. It follows that k = 1 and G ≤ B = T1 . Suppose that B > G. Now |B : G| = |Bv : G v | = |Bv |/24, and it follows from [4, Theorem 3.9] and [5, Theorem] that |Bv | divides 25 32 or 24 36 . Hence |B : G| divides 22 3 or 2.35 , that is, B has a subgroup G of index dividing m where m = 22 3 or 2.35 . Let Q be a maximal subgroup of B which contains G. Then |B : Q| divides m and B has a primitive permutation representation on [B : Q] with Q the stabilizer of a point. Note that m ≤ 486, p ≥ 23 and Q contains G (∼ = PSL(2, p)). Checking [2, Appendix B], there exists no such group B, which is a contradiction. Thus B = G, and so A = PSL(2, p) or PGL(2, p), which is not possible since σ ∈ A and σ centralizes G. Therefore, A is not quasiprimitive on V 0. Let R be a maximal normal subgroup of A which is intransitive on V 0. Let 0 R be the quotient graph of 0 induced by R (see [9, p. 233]). Then A := A/R induces a quasiprimitive permutation group on V 0 R , and A ≤ Aut 0 R . As G is a simple group, 0 is non-bipartite. Therefore, 0 R is of valency 4 and G/R is quasiprimitive, see [9, Theorem 4.1]. Since G is simple, G ∼ = G = G R/R ≤ A. Arguing as in the previous paragraph (replacing (G, A) by (G, A)), we can obtain that G = soc(A) and so A = PSL(2, p) or PGL(2, p). Suppose that A > G. Then A ∼ = PGL(2, p). Since 0 R is of valency 4, both G v and Av are {2, 3}-groups, where v is a vertex of 0 R . Since G v = H ∼ = A4 and R 6= 1, it follows from the list (∗) that G v ∼ = S4 . Now Av > G v ∼ = S4 . However, it is known (see [12, pp. 416– 418]) that PGL(2, p) does not have a subgroup that is a {2, 3}-group and contains a proper subgroup isomorphic to S4 , which is a contradiction. Therefore, A = G ∼ = PSL(2, p). Since 0 is (A, 2)-arc transitive, by [9, Theorem 4.1], R is semiregular on V 0 so that |R| = |V 0|/|V 0 R | = |G v |/|G v | = 2. Consequently, R = hσ i ∼ = Z2 and A = G × hσ i. 2 A CKNOWLEDGEMENTS The author is very grateful to Cheryl E. Praeger who introduced the topic to him and gave many helpful suggestions on this work, and acknowledges support of an ARC grant.
R EFERENCES 1. — R. W. Baddeley, Two-arc transitive graphs and twisted wreath products, J. Algebraic Combin. 2 (1993), 215–237. 2. — J. D. Dixon and B. Mortimer, Permutation Groups, Springer, Hong Kong, 1996. 3. — X. G. Fang and C. E. Praeger, Finite two-arc transitive graphs admitting a Suzuki simple group, Research Report, Sep 1996/26, UWA. 4. — A. Gardiner, Arc transitivity in graphs, Quart. J. Math. Oxford Ser (2) 24 (1973), 399–407. 5. — A. Gardiner, Arc transitivity in graphs II, Quart. J. Math. Oxford Ser (2) 25 (1974), 163–167. 6. — A. Hassani, L. Nochefranca and C. E. Praeger, Two arc transitive graphs admitting a twodimensional projective linear group, preprint (1995).
502
Cai Heng Li
7. — A. A. Ivanov and C. E. Praeger, On affine 2-arc transitive graphs, Eur. J. Comb. 14 (1993), 421–444. 8. — C. H. Li, On finite groups with the Cayley invariant property, Bull. Austral. Math. Soc. 56 (1997), 253–261. 9. — C. E. Praeger, An O’Nan–Scott theorem for finite quasiprimitive groups and an application to 2-arc transitive graphs, J. London Math. Soc. 47 (1993), 227–239. 10. — C. E. Praeger, Finite transitive permutation groups and finite vertex-transitive graphs, Graph Symmetry: Algebraic Methods and Applications, NATOASI Ser. C 497, pp. 277–318. 11. — C. E. Praeger, Finite quasiprimitive groups, Surveys in Combinatorics, London Mathematic Society, Lecture Note Series 241 (1977), 65–86. 12. — M. Suzuki, Group Theory I, Springer, Berlin, 1982. Received 17 December 1996 and accepted 13 October 1997 C AI H ENG L I Department of Mathematics, The University of Western Australia, Perth, WA 6907, Australia
A Family of Quasiprimitive 2-arc Transitive Graphs which Have Non-quasiprimitive Full Automorphism Groups C AI H ENG L I For each prime p ≡ 1 or −1(mod 24), we construct a (PSL(2, p), 2)-arc transitive graph 0 of valency 4 such that Aut 0 = PSL(2, p) × Z 2 . Thus there exists an infinite family of quasiprimitive 2-arc transitive graphs which have non-quasiprimitive full automorphism groups. c 1998 Academic Press
Let 0 = (V 0, E0) be a finite connected graph with vertex set V 0 and edge set E0. For a positive integer s, an s-arc of 0 is an (s + 1)-tuple (v0 , v1 , . . . , vs ) of vertices such that (vi−1 , vi+1 ) ∈ E0 for 1 ≤ i ≤ s and vi−1 6= vi+1 for 1 ≤ i ≤ s − 1. The graph 0 is said to be (G, s)-arc transitive if G ≤ Aut 0 and G is transitive on V 0 and on the set of s-arcs of 0. For an intransitive normal subgroup N of G, the quotient graph 0 N induced by N is defined as the graph with vertices the N -orbits on V 0 such that two N -orbits 11 and 12 are adjacent in 0 N if some vertex of 11 is adjacent in 0 to some vertex of 12 . It is shown in [9, Theorem 4.1] that if 0 is a non-bipartite (G, 2)-arc transitive graph then 0 N is a (G/N , 2)-arc transitive graph. Thus 0 is a cover of 0 N . If N is a maximal intransitive normal subgroup of G, then every non-trivial normal subgroup of G/N is transitive on V 0N . Let G be a transitive permutation group on . If there exists no non-trivial G-invariant partition of , then G is said to be primitive on ; if every non-trivial normal subgroup of G is transitive on then G is said to be quasiprimitive. Thus a primitive group is quasiprimitive, but the converse is not necessarily true, see [10]. Praeger [9] obtained a version of the O’Nan–Scott theorem for quasiprimitive groups, which divides quasiprimitive groups into eight classes, and then she proved that only four of them can act 2-arc transitively on a graph. This suggests a reasonable program to describe non-bipartite 2-arc transitive graphs by two steps: (i) describing graphs admitting a quasiprimitive 2-arc transitive group of automorphisms, (ii) characterizing 2-arc transitive covers of examples from (i). There has been some work related to this program, see for example [1, 3, 6, 7]. To determine isomorphism classes of quasiprimitive 2-arc transitive graphs, it is necessary to determine the full automorphism group of a quasiprimitive 2-arc transitive graph. However, this is a very difficult problem. In this note we investigate a question relevant to this problem. For a (G, 2)-arc transitive graph 0, G ≤ Aut 0 ≤ Sym(V 0), where Sym(V 0) is the symmetric group on V 0. If G is primitive then Aut 0 must be primitive. Thus a natural question arises (refer to Baddeley [1, Section 6] or Praeger [11, Section 7]): if G is quasiprimitive on V 0, is Aut 0 quasiprimitive? Baddeley [1, Section 6] constructed a single counterexample which is a graph with a group of twisted-wreath-action type (in terms of [9]) such that there exists a non-quasiprimitive group H satisfying G < H ≤ Aut0. In this paper we give the first infinite family of counterexamples such that Aut0 is not quasiprimitive. 0195-6698/98/040499 + 04
$25.00/0
c 1998 Academic Press
500
Cai Heng Li
For a group G and a core-free subgroup H of G, if there exists an element g ∈ G such that g 2 ∈ H and G = hH, gi, then we have a connected G-arc transitive graph 0 := 0(G, H, g) with V 0 = {H x | x ∈ G} and E0 = {{H x, H y} | x y −1 ∈ H g H }. Denote by [H : H ∩ H g ] the set of cosets of H ∩ H g in H . Then 0 is (G, 2)-arc transitive iff H acts 2-transitively on [H : H ∩ H g ] (see for example [3, Theorem 2.1]). The result of this paper is the following theorem. T HEOREM . Let p be a prime such that p ≡ 1 or −1 (mod 24). Let G = PSL(2, p). Then there exists a (G, 2)-arc transitive graph 0 of valency 4 such that G is quasiprimitive on V 0 while Aut 0 = G × Z2 and in particular Aut 0 is not quasiprimitive on V 0. P ROOF. Since p ≡ ε = ±1 (mod 24), p ≥ 23. Then by [12, p. 417], each maximal subgroup of G is isomorphic to one of the following: S4 , A5 , D p−1 , D p+1 , Z p ×| Z p−1 . 2
Note that |G| = p( p + 1)( p − 1)/2, and ( p − 1, p + 1) = 2. It follows that a Sylow 3subgroup G 3 of G is conjugate to a subgroup of D p−ε . Thus G 3 is cyclic and all subgroups of G of order 3 are conjugate. Let M be a subgroup of G isomorphic to S4 , and let H be a subgroup of M isomorphic to A4 . We shall construct a class of (G, 2)-arc transitive graphs of valency 4 with H the stabilizer of a point. (Recently, the author became aware that this class of graphs has been constructed in [6]. However, for the sake of completeness and in need for notation in the ensuing, we present a construction here.) By the discussion before the theorem it is sufficient to find an element g ∈ G such that hH, gi = G, g 2 ∈ H , |[H : H ∩ H g ]| = 4 and H is 2-transitive on [H : H ∩ H g ]. Now M and H may be written as M = (N ×| L)×| hbi and H = N ×| L, where N ∼ = Z22 , ∼ ∼ ∼ L = Z3 and hbi = Z2 such that hL , bi = S3 . Consider NG (L). By our observation above, NG (L) ∼ = D p−ε . Consequently, NG (L) can be written as NG (L) = (L 0 × hci)×| hbi for some c ∈ G, where L ≤ L 0 , L 0 is a Sylow 3-subgroup of G, and |L 0 |o(c) = ( p − ε)/2. Let g be the involution of hci. Then g centralizes both L 0 and b. Suppose that hH, gi < G. Then there exists a maximal subgroup M0 of G containing hH, gi. Since g ∈ / H , hH, gi > H ∼ = S4 or A5 . = A4 . It follows from the list (∗) that M0 ∼ However, since C M0 (L) ≥ ChH,gi (L) ≥ L × hgi ∼ = Z6 , we have that M0 6 ∼ = S4 , A5 , which is a contradiction. Therefore, hH, gi = G. 2 g Suppose that N0 := N ∩ N g 6= 1. Since g 2 = 1, N0 = N g ∩ N g = N0 . So g normalizes hL , b, N0 i. Since L is transitive on the non-identity elements of N by conjugation, hL , b, N0 i = M, and hence g normalizes M, which is a contradiction since hM, gi = G is simple. Therefore, N ∩ N g = 1 and so H ∩ H g = L. It follows that |[H : H ∩ H g ]| = 4 and H is 2-transitive on [H : H ∩ H g ]. Let 0 = 0(G, H, g). Since g 2 = 1, g 2 ∈ H . Thus 0 is a connected (G, 2)-arc transitive graph (by the discussion before the theorem) and G is quasiprimitive on V 0 (since G is simple). Set F := G × hσ i where hσ i ∼ = S4 and = Z2 . Let K = (N ×| L)×| hbσ i. Then K ∼ hK , gi = hH, bσ, gi = hG, bσ i = F. Since g centralizes both b and σ , g centralizes bσ , and further, since g centralizes L, g normalizes hL , bσ i. Since N ∩ N g = 1, K ∩ K g = hL , bσ i ∼ = S3 . It follows that |[K : K ∩ K g ]| = 4 and K is 2-transitive on [K : K ∩ K g ]. Thus 6 := 0(F, K , g) is a connected (F, 2)-arc transitive graph. Let v be the vertex of 6 corresponding to K . Since G < F and G v = Fv ∩ G = K ∩ G = N ×| L = H , it follows that 6 = 0. Since hσ i F, F is not quasiprimitive on V 0.
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Quasiprimitive 2-arc transitive graphs
501
Finally, we prove that Aut 0 = F = G × hσ i so that Aut 0 is not quasiprimitive on V 0. Let A = Aut 0. Then F ≤ A and A is a transitive permutation group on V 0. Since 0 is of valency 4, every prime divisor is not greater than 4 so that Av is a {2, 3}-group (see [8, Lemma 2.1]). Suppose that A is quasiprimitive on V 0. By [9, Theorem 1.1], B := soc(A) = T1 × · · · × Tk where k ≥ 1 and T1 ∼ = ··· ∼ = Tk is simple. Since B A, we have G ∩ B G. Since A is quasiprimitive on V 0, B is transitive on V 0 and C A (B) ≤ B. Therefore, G ≤ B. Further, since T1 B, G∩T1 G, and so either G∩T1 = 1 or G∩T1 = G. Since Av is a {2, 3}-group, Bv is a {2, 3}-group. If G ∩ T1 = 1 then |hG, T1 i| = |G||T1 | dividing |A|, which is a contradiction since |A|/|G| = 2r 3s for some r, s ≥ 0. It follows that k = 1 and G ≤ B = T1 . Suppose that B > G. Now |B : G| = |Bv : G v | = |Bv |/24, and it follows from [4, Theorem 3.9] and [5, Theorem] that |Bv | divides 25 32 or 24 36 . Hence |B : G| divides 22 3 or 2.35 , that is, B has a subgroup G of index dividing m where m = 22 3 or 2.35 . Let Q be a maximal subgroup of B which contains G. Then |B : Q| divides m and B has a primitive permutation representation on [B : Q] with Q the stabilizer of a point. Note that m ≤ 486, p ≥ 23 and Q contains G (∼ = PSL(2, p)). Checking [2, Appendix B], there exists no such group B, which is a contradiction. Thus B = G, and so A = PSL(2, p) or PGL(2, p), which is not possible since σ ∈ A and σ centralizes G. Therefore, A is not quasiprimitive on V 0. Let R be a maximal normal subgroup of A which is intransitive on V 0. Let 0 R be the quotient graph of 0 induced by R (see [9, p. 233]). Then A := A/R induces a quasiprimitive permutation group on V 0 R , and A ≤ Aut 0 R . As G is a simple group, 0 is non-bipartite. Therefore, 0 R is of valency 4 and G/R is quasiprimitive, see [9, Theorem 4.1]. Since G is simple, G ∼ = G = G R/R ≤ A. Arguing as in the previous paragraph (replacing (G, A) by (G, A)), we can obtain that G = soc(A) and so A = PSL(2, p) or PGL(2, p). Suppose that A > G. Then A ∼ = PGL(2, p). Since 0 R is of valency 4, both G v and Av are {2, 3}-groups, where v is a vertex of 0 R . Since G v = H ∼ = A4 and R 6= 1, it follows from the list (∗) that G v ∼ = S4 . Now Av > G v ∼ = S4 . However, it is known (see [12, pp. 416– 418]) that PGL(2, p) does not have a subgroup that is a {2, 3}-group and contains a proper subgroup isomorphic to S4 , which is a contradiction. Therefore, A = G ∼ = PSL(2, p). Since 0 is (A, 2)-arc transitive, by [9, Theorem 4.1], R is semiregular on V 0 so that |R| = |V 0|/|V 0 R | = |G v |/|G v | = 2. Consequently, R = hσ i ∼ = Z2 and A = G × hσ i. 2 A CKNOWLEDGEMENTS The author is very grateful to Cheryl E. Praeger who introduced the topic to him and gave many helpful suggestions on this work, and acknowledges support of an ARC grant.
R EFERENCES 1. — R. W. Baddeley, Two-arc transitive graphs and twisted wreath products, J. Algebraic Combin. 2 (1993), 215–237. 2. — J. D. Dixon and B. Mortimer, Permutation Groups, Springer, Hong Kong, 1996. 3. — X. G. Fang and C. E. Praeger, Finite two-arc transitive graphs admitting a Suzuki simple group, Research Report, Sep 1996/26, UWA. 4. — A. Gardiner, Arc transitivity in graphs, Quart. J. Math. Oxford Ser (2) 24 (1973), 399–407. 5. — A. Gardiner, Arc transitivity in graphs II, Quart. J. Math. Oxford Ser (2) 25 (1974), 163–167. 6. — A. Hassani, L. Nochefranca and C. E. Praeger, Two arc transitive graphs admitting a twodimensional projective linear group, preprint (1995).
502
Cai Heng Li
7. — A. A. Ivanov and C. E. Praeger, On affine 2-arc transitive graphs, Eur. J. Comb. 14 (1993), 421–444. 8. — C. H. Li, On finite groups with the Cayley invariant property, Bull. Austral. Math. Soc. 56 (1997), 253–261. 9. — C. E. Praeger, An O’Nan–Scott theorem for finite quasiprimitive groups and an application to 2-arc transitive graphs, J. London Math. Soc. 47 (1993), 227–239. 10. — C. E. Praeger, Finite transitive permutation groups and finite vertex-transitive graphs, Graph Symmetry: Algebraic Methods and Applications, NATOASI Ser. C 497, pp. 277–318. 11. — C. E. Praeger, Finite quasiprimitive groups, Surveys in Combinatorics, London Mathematic Society, Lecture Note Series 241 (1977), 65–86. 12. — M. Suzuki, Group Theory I, Springer, Berlin, 1982. Received 17 December 1996 and accepted 13 October 1997 C AI H ENG L I Department of Mathematics, The University of Western Australia, Perth, WA 6907, Australia