Scott Theorem for finite primitive permutation groups. It is shown that ..... may be interpreted as the additive group of a A:-dimensional vector space over GF(/?).
AN O'NAN-SCOTT THEOREM FOR FINITE QUASIPRIMITIVE PERMUTATION GROUPS AND AN APPLICATION TO 2-ARC TRANSITIVE GRAPHS CHERYL E. PRAEGER
ABSTRACT A permutation group is said to be quasiprimitive if each of its nontrivial normal subgroups is transitive. A structure theorem for finite quasiprimitive permutation groups is proved, along the lines of the O'NanScott Theorem for finite primitive permutation groups. It is shown that every finite, non-bipartite, 2-arc
transitive graph is a cover of a quasiprimitive 2-arc transitive graph. The structure theorem for quasiprimitive groups is used to investigate the structure of quasiprimitive 2-arc transitive graphs, and a new construction is given for a family of such graphs.
1. Introduction The motivation for this paper came from a study of .s-arc transitive graphs, 5 ^ 2 . An s-arc, s ^ 1, in a graph F is a sequence (a 0 ,..., as) of s + 1 vertices of T such that, for all 1 ^ / < s, 2). Since Ga is transitive on the set of simple direct factors of/, Ga is transitive on {D1,...,Dl} and hence each Dt involves precisely m of the factors T(, so k = Im. Let P be the permutation group induced by G on the set 3~ = {Tx,..., 7^.}, of simple direct factors of M. Assume first that / = 1. If CG(J) # 1 then we must have k = 2, J ~ CG(J) ~ T, and G is primitive on Q. If on the other hand CG(J) = 1 so that M = J, then P is transitive. If in this case P preserves a nontrivial partition of 2T then the subgroup Y of all elements of M constant on each block of the partition is Ga-invariant and we have Ga< GaY < G whence G is not primitive on Q. If on the other hand P is primitive on 2T then it is straightforward to show that Ma is a maximal proper Ga-invariant subgroup of M, and hence that Ga is maximal in G, that is G is primitive on Q. Thus G is primitive on Q. if and only if either k = 2 and P = 1, or P is primitive on ^". The group G is shown to be of type III(a) exactly as in [5, p. 393]. Now let / > 1 and set K = Tx x ... x Tm and N = NG(K). The set {7;,..., TJ is a block of imprimitivity for the action of G on 3T and N is its setwise stabilizer. Thus if M = J then N is transitive on {Tx,..., Tm} and K is a minimal normal subgroup of N, while if CG(J) # 1 then JV has two orbits in {7;,..., TJ and K is the direct product K = (J (] K)x (CG(J) n K) of two minimal normal subgroups of N. In the latter case Ka is a diagonal subgroup of K = (J n AT) x (C G (/) n .K) since Ma is a diagonal subgroup of Jx CG(J); also Ka = D1~Tand it follows in this case that J 0 K~ CG(J) n K~ 7 a n d m = 2. For L ^ N denote by L* the group of automorphisms of K induced by L, so that L* = LCG(K)/CG(K). Since TV contains M, we see that Af is transitive on Q and so N = MNa. Hence W* = K*N*. Set H = N* and let T be the coset space (H:N*). Then /7has socle K* ~ A^and either AT* is a minimal normal subgroup of H and is transitive on T, or K* is the direct product of two minimal normal subgroups of H, (/ n K)* and (CG(J) D AT)*, each of which is isomorphic to T and is regular on F. Thus H is a quasiprimitive permutation group on F of type III(a). Also |Q| = |F|'. We now claim that G n is permutationally equivalent to a subgroup of HwrSt in its natural product action on F', hence to a group of type III(b)(ii). We saw above that Ga is transitive on {D1,...,Dl}. Let 0t = {g 1? ...,g t } be a right transversal for Na in Ga and hence for N in G, such that D^ = Di for 1 < / < /. Write Kx = K°< for 1 ^ i ^ /, so that G permutes the set {Kx,...,Kt} transitively. For geG, write g = ngg with and ngeN. Writing elements of Hv/rSt in the form {h1,...,h^)n with hteH, i and n permuting the components hi naturally, we define a map p.G -* HwrSl by
p:gh+(a*,...,a*)n
(geG),
where n is the permutation induced by g on {A"15..., Kt}, for 1 ^ ir ^ / we have a i = SiSiSiST1-: a n d a* denotes the automorphism of A" induced by conjugation by av Then p is a monomorphism; moreover, since gt € Ga for all /, we have Gap ^ A7'* wr Sj, the point stabilizer in the natural action of H wr S1, on F'. Since |Q| = |F|', identification of G with its image Gp gives the required embedding of G in / / w r S , , acting naturally on F'. This proves our claim. Case 2. Now suppose that Rt = n^MJ is a proper subgroup of Tf for each / = 1,..., k. Then Ma is certainly not a diagonal subgroup of / x CG(J) and hence
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CG(J) = 1. Thus M = J is a minimal normal subgroup of G. In particular G = MG3, and hence Ga is transitive on {Tx,..., Tk}. Thus for / = \,...,k,Rt is the image of Rx under an isomorphism 7] -> 7J. Since R = Rxx ...xRk is Ga-invariant we have Ga^GaR < G. Thus G preserves a partition of Q. such that the setwise stabilizer of the block containing a is GaR, and G acts faithfully on this partition. Note that GaR n M = R. In particular if M a = 1 then R = 1 and the partition consists of parts of size 1. Thus if M a = 1 we may identify Q with this partition. Set N = JV G (7[) and for L ^ N denote by L* the group of automorphisms of Tx induced by L by conjugation, so that L* = LCG(7^)/CG(7]). Since N contains the transitive subgroup M we have N = MNa. Hence N* ^ T*. Case 2(a). Suppose that Rx = 1 and hence that Ma = 1. Define :Na^> Aut Tx to be the natural homomorphism (that is, for neNa and teTv(n):ti->tn), so that Ker0 = CG{TX) n Ga. Write Z = 0- x (Inn TJ. Then since CG(M) = I, core G (Z) = 1. Set P = Ga and Q = Nx so that P acts transitively on {Tlt..., 7;.} and G = MP.'Abusing notation slightly, take P to act on 7 = {l,...,fc} by Tf = Tip (iel, peP). Then as in [5, p. 395] the map 6 defined below is an isomorphism of G onto the twisted wreath product Tx twr^ (P1) (defined in Section 2) which maps M onto the base group B and Ga = P onto the top group P\ so that G is of type III(c). To define 6 choose c(eP such that 77' = 7^ for / = 1,...,k so that {c 15 ...,c k ) is a transversal for Q in / \ Now each m e M is of the form m = f|f=1 a{ with a, e 7^ and hence a^ e 7^ for / = 1,..., k. Now 6: G -*• Tx twr^ P is defined by for m = P[ at e M and u e P, where # m : P ->• 7] is the map given by 0TO(c< ^) = a\iq for / = 1 ,...,k and ^r e 2- Clearly Bm belongs to the base group B of Tx twr^ P', and ^ is 1 — 1 and hence bijective. Showing that 6 is a homomorphism is equivalent to showing, for all m,m'EM and ueP, that 0m.mu = 9m-(6m)u, and this is equivalent to showing that
6nu = (6my for m = Y[aiEM,ueP. The details of this calculation are given in [5, p. 395]. Thus 0 is an isomorphism and, since 0(M) = B and 0(Ga) = P it follows that G is of type III(c). Case 2{b). Thus we may assume that Rx^ \. We shall show that T* ^ iV*. Assume to the contrary that7\* ^ N*. Then N* = T* N* = N*, that is N = Na C^TJ. Since Rx # 1, it follows that 7; = > ^ = = ^ Ga7? which is not the case as GaR n M = R does not contain 7^. Hence T* ^ N*, and in fact L* := Af* D T* has a normal subgroup R[* ~ Rx. Let Li be the subgroup of Tx containing Rx such that Lf = L*. Let L ; be the image of Lx under conjugation by an element of Ga which takes Tx to Tt, so that Rt^L{< T{, for / = 2,...,k, and let L:= Lx x ... x L t . Further let H = N* and let T be the coset space (H:N*). Then / / has socle T* ~ 7^ and so H is quasiprimitive on T of type II. Also |r| = | 7^:7^1 and the coset space A = (G:G a L) has size |A| = \M:L\ = |F|fc. A calculation along the lines of that for the case / > 1 of Case I shows that GA is permutation equivalent to a subgroup of HwrSt in its product action on rk. Moreover G cz GA, and so G is of type III(b)(i). This completes the proof of Theorem 1.
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4. s-arc transitive graphs, s ^ 2 Let F be a finite connected graph and let G ^ Aut F act transitively on the set of 5-arcs of F, where s > 2. We shall show that, if F is not bipartite, then F is a cover of a finite connected graph I whose automorphism group contains a subgroup which is quasiprimitive on the vertices and transitive on the s-arcs of Z. A graph F will be called a cover of a graph I if there is a surjection 0 from the vertex set of F to the vertex set of I which preserves adjacency and is such that, for each vertex a of F, the sets F^a) and ^ ( a ^ ) of neighbours of a and a0 in F and 2 respectively have the same size. First we investigate the action of a normal subgroup N of G on the vertices of F. If TV is intransitive on vertices then the quotient graph TN is defined as the graph with vertices the TV-orbits such that two TV-orbits A1 and A2 are adjacent in F v if some vertex of At is adjacent in F to some vertex of A2. In the case where TV has two orbits, Ax and A2, the graph F is bipartite, Ax and A2 are the two parts of the bipartition, and TN = K2 is the complete graph on two vertices. The case where TV has more than two orbits is more interesting for in this case F is a cover of F^ and F ^ is s-arc transitive. THEOREM 4.1. Let G be a group of automorphisms of a finite connected graph T such that G is s-arc transitive on F, for some s ^ 2. Suppose that G has a normal subgroup TV which has more than two orbits on vertices. Then TN is finite and connected and the group of automorphisms of TN induced by G is s-arc transitive on TN. Also TV is semiregular on vertices {that is Na= 1 for each vertex . The group G£l(a) is doubly transitive, since Ga is transitive on the set of 2-arcs (oc^a^a,,) with OLX = a, and consequently N^{a) is transitive. It follows that every neighbour of a lies in the single TV-orbit A1 containing F^a). Hence every neighbour of every vertex of the TV-orbit A2 = 2 (as T is not a cycle) it follows that x = x~l, that is x2 = 1. So X is a Ga-class of involutions and the fact that = M follows from the connectivity of Y. A further discussion of graphs with G of type III(c) is contained in [7, Section 3.4]. An example of such a graph with G primitive on vertices was found by Robert Baddeley [2]. We give below a general construction of quasiprimitive 2-arc transitive Cayley graphs with a group of automorphisms of type III(c). CONSTRUCTION 6.3. Let k and / be positive integers such that (a) there is a doubly transitive permutation group P on a set £ which has a corefree subgroup Q of index k with / orbits in Z, Q not the stabilizer of a point of E, and (b) there is a nonabelian simple group T such that T is generated by certain of its (The yi need not be distinct.) involutions y^...,yv (Recall that Q is core-free in P if f]gePQ9 = {1P}.) Then P acts faithfully and transitively by right multiplication on the set of k right cosets of Q in P. Thus we may consider P acting on M = Tk = 7[ x ... x Tk by permuting the simple direct factors Z[,..., Tk of M as it permutes the cosets of Q. Then the semidirect product G = MP is just the permutational wreath product Twr P. Let a el,. Note that, as Q has / orbits in Z, the stabilizer Po has / orbits on the set of simple direct factors of M. Let these Pff-orbits be denoted 6l,...,dl. We shall define a Cayley graph on M which admits G as a quasiprimitive 2-arc transitive group of automorphisms of type III(c). Define x — (x1,...,xk) to be the element of M with ith entry xt = y} where TJefy, for each i = \,...,k. Set X = {x°\geP}, and define T to be the graph with vertex set M such that m,m' are adjacent if and only if mm''1 e X.
LEMMA 6.4. The graph Y defined in Construction 6.3 admits G as a quasiprimitive, 2-arc transitive group of automorphisms. Proof. It follows from the construction that Y admits M acting by right multiplication, and P acting by conjugation (as in G) and hence Y admits G = MP as a group of automorphisms. Moreover G is quasiprimitive on vertices as M is the unique minimal normal subgroup of G and M is transitive. To show that G is transitive on the 2-arcs of Y it is sufficient to show that Ga = P (where a = 1M) is doubly transitive on I\(a) = X. Now by the definition of x e X it is clear that Pa fixes
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x and, as P is doubly transitive on S, Pa is a maximal subgroup of P (and P does not fix x). It follows that G^ = Pa. Thus the action of Ga on F^a) is equivalent to the action of P on I, and hence is doubly transitive. Sometimes a graph obtained by Construction 6.3 will be disconnected. Requiring Q to be a maximal subgroup of P is sufficient to ensure that F is connected. LEMMA
6.5. If Q is maximal in P then F is connected.
Proof. The connected component of F containing a = \M is L = . Since (^j,.,. j , ) = !Tand since P is transitive on the simple direct factors on M, it follows that L projects onto each of Tx,..., Tk. Thus (see [10, p. 328]) L is a direct product of, say, a copies of T, each a diagonal subgroup of T6 where ab = k. Then P must preserve the partition of {Tv..., 7^} into b sets of size a determined by this direct product L. However Q maximal in P is equivalent to the action of P on {7^, ...,7^.} being primitive. Thus either a = k, and the lemma is proved, or a = 1. Suppose that a = 1. Then L ~ T. To show that this cannot happen we use the fact that Q is not the stabilizer of a point in E. Since Q ^ Pa we have some geQ with a" ^ cr, and hence x° ^ x (since i^ = G^). On the other hand Q is the stabilizer of one of the simple direct factors of M, say Tx, and hence x9 and x have the same first entry x15 an involution of T. Thus we have xx°, a nonidentity element of L withfirstentry 1T. so L is not isomorphic to T. Thus Lemma 6.5 is proved. It is certainly not necessary for P to act primitively on the simple direct factors of M to get a connected graph P. (There is an example with T = Ab, k = 10, / = 3, and P a Frobenius group of order 20.) References 1. L. BABAI, 'Arc transitive covering digraphs and their eigenvalues', J. Graph Theory 9 (1985) 363-370. 2. R. BADDELEY, 'Two-arc transitive graphs and twisted wreath products', preprint, 1992. 3. P. J. CAMERON, 'Finite permutation groups and finite simple groups', Bull. London Math. Soc. 13 (1981) 1-22. 4. A. A. IVANOV and C. E. PRAEGER, 'On finite affine 2-arc transitive graphs', European J. Combin. to appear. 5. M. W. LIEBECK, C. E. PRAEGER and J. SAXL, 'On the O'Nan-Scott Theorem for finite primitive permutation groups', J. Austral. Math. Soc. Ser. A 44 (1988) 389-396. 6. B. H. NEUMANN, 'Twisted wreath products of groups', Arch. Math. 14 (1963) 1-6. 7. C. E. PRAEGER, 'Primitive permutation groups with a doubly transitive subconstituent', J. Austral. Math. Soc. Ser. A 45 (1988) 66-77. 8. C. E. PRAEGER, 'Finite vertex transitive graphs and primitive permutation groups', to appear in the Proceedings of the Marshall Hall Conference, 1990. 9. C. E. PRAEGER, 'On a reduction theorem for finite, bipartite, 2-arc transitive graphs', Australas. J. Combin. to appear. 10. L. L. SCOTT, 'Representations in characteristic p\ Santa Cruz Conference on finite groups, Proc. Sympos. Pure Math. 37 (1980) 318-331. 11. M. SUZUKI, Group Theory /(Springer, Berlin, 1982). 12. R. M. WEISS, 'A characterization of the group M12\ Algebras Groups Geom. 2 (1985) 555-563.
Department of Mathematics University of Western Australia Nedlands Western Australia 6009 Australia
