ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 13 (2017) 137-165 Affine primitive symmetric graphs of diameter two* * Carmen Amarra t Institute of Mathematics, University of the Philippines Diliman C. P. Garcia Avenue, Diliman, Quezon City 1101, Philippines Michael Giudici, Cheryl E. Praeger Centre for the Mathematics ofSymmetry and Computation, The University ofWestern Australia 35 Stirling Highway, Perth, WA 6009, Australia Received 28 January 2016, accepted 15 May 2016, published online 22 February 2017 Let n be a positive integer, q be a prime power, and V be a vector space of dimension n over Fq. Let G := V x G0, where G0 is an irreducible subgroup of GL (V) which is maximal by inclusion with respect to being intransitive on the set of nonzero vectors. We are interested in the class of all diameter two graphs r that admit such a group G as an arc-transitive, vertex-quasiprimitive subgroup of automorphisms. In particular, we consider those graphs for which G0 is a subgroup of either rL(n, q) or TSp(n, q) and is maximal in one of the Aschbacher classes Cj, where i G {2,4,5,6,7, 8}. We are able to determine all graphs r which arise from G0 < rL(n, q) with i G {2,4,8}, and from G0 < TSp(n, q) with i G {2,8}. For the remaining classes we give necessary conditions in order for r to have diameter two, and in some special subcases determine all G-symmetric diameter two graphs. Keywords: Symmetric graphs, Cayley graphs, quasiprimitive permutation groups, linear groups. Math Subj. Class.: 05C25, 20B15, 20B25 *This paper forms part of the first author's Ph.D., which is supported by an Endeavour International Postgraduate Research Scholarship (with UPAIS) and a Samaha Top-Up Scholarship from The University of Western Australia, and forms part of the Australian Research Council Discovery project DP0770915 held by the last two authors. t Corresponding author. E-mail address: mcamarra@math.upd.edu.ph (Carmen Amarra), michael.giudici@uwa.edu.au (Michael Giudici), cheryl.praeger@uwa.edu.au (Cheryl E. Praeger) Abstract ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 138 Ars Math. Contemp. 13 (2017) 107-123 1 Introduction A symmetric graph is one which admits a subgroup of automorphisms that acts transitively on its arc set; if G is such a subgroup, we say in particular that the graph is G-symmetric. We are interested in the family of all symmetric graphs with diameter two, a family which contains all symmetric strongly regular graphs. We consider those G-symmetric diameter two graphs where G is a primitive group of affine type, and where the point stabiliser G0 is maximal in the general semilinear group or in the symplectic semisimilarity group. Our main result is Theorem 1.1. Those affine examples where G0 is not contained in either of these groups were studied in [2]. Theorem 1.1. Let V = F^ for some prime power q and positive integer n, and let G = V x G0, where G0 is an irreducible subgroup of the general semilinear group rL(n, q) or the symplectic semisimilarity group TSp(n, q), and G0 is maximal by inclusion with respect to being intransitive on the set of nonzero vectors in V. If r is a connected graph with diameter two which admits G as a symmetric group of automorphisms, then r is isomorphic to a Cayley graph Cay(V, S) for some orbit S of G0 satisfying (S) = V and S = -S, and one of the following holds: 1. (Go, S) are as in Tables 1.0.1 and 1.0.2; 2. G0 satisfies the conditions in Table 1.0.3; 3. G0 belongs to the class C9. Furthermore, all pairs (G0, S) in Tables 1.0.1 and 1.0.2 yield G-symmetric diameter two graphs Cay(V, S). Notation for Tables 1.0.1 and 1.0.2. The set Xs is as in (3.2) and Wg is as in (3.5) in Section 3.2, Ys is as in (3.7) in Section 3.3, c(v) is as in (3.9) in Section 3.4, S0 is as in (2.4) in Section 2.2, and S#, So and SK are as in (3.1) in Section 3.1. Cayley graphs are defined in Section 2.1. The graphs marked f did not appear in [2]. Table 1.0.1: Symmetric diameter two graphs from maximal subgroups of rL(n, q) Go n GL (n,q) S Conditions 1 GL (m, q) i Sym (t), mt = n Xs qm > 2 and s > t/2 2 GL (k, q) 1 min {fc, m} ts GL (n, q1/r) o Z?_i, r > 2 and n > 2 vGo as in (3.14) c(v) = r — 1 or c(v) = r t4 GL in, q1/r J o Z„_i, r = 2 or n = 2 vGo as in (3.14) c(v) = 1 5 (Zq_i o (Z4 o Qs)).Sp (2, 2), n = 2, q odd vGo v e V * 6 GL (m, q) i® Sym (2), m2 = n Ys s > m/2 t7 GU (n, q), n > 2 So, S# 8 GO (n, q), n = 3 and q = 3 So 9 GO (n, q), nq odd, n > 3 or q > 3 So, So, or SH 10 GO+ (n, q), n even, q odd, n > 2 or q > 2 So or S# 11 GO_ (n, q), n even, q odd, n > 2 So or S# C. Amarra et al.: Affine primitive symmetric graphs of diameter two 139 Table 1.0.2: Symmetric diameter two graphs from maximal subgroups of TSp(n, q) Go n GL (n, q) S Conditions 1 t2 ts 4 5 6 CTeAut(F, ) Go Sp (m, q)4 .[q - 1].Sym (t), mt = n Xs GL (m,q) .[2], 2m = n |J. (Zq—i o Qs).O- (2, 2), n = 2, q odd v GO+(n,q), n = 2 and q = 2 So GO+ (n, q), q and n even, n > 2 or q > 2 S0 or S# GO— (n, q), q and n even, n > 2 S0 or S# qm > 2 and s > t/2 Wß* qm > 2 and ß e Fq v e V * Table 1.0.3: Restrictions for remaining cases Go n GL (n,q) Conditions Restrictions 1 GSp (k, q) ®GO£(m,q), m odd, q> 3 GL (n,q1/r) o Z, q-1 S (Zq—i o R).Sp (2t,r), n = r4 4 (Zq—1 o R).Sp (2t, 2), n = r4 5 (Zq—1 o R).O- (2t, 2), n = r4 6 GL (m, q) i® Sym (t), m4 = n 7 GSp (m,q) i® Sym (t), m4 = n, q odd t > 3 Proposition 3.14 c(v) = r - 1, r Proposition 3.16 (2), (3), (4) R Type 1, t > 2 Proposition 3.23 (1) R Type 2, t > 2 Proposition 3.23 (2) R Type 4, t > 2 Proposition 3.23 (3) t3 Proposition 3.25 Proposition 3.26 2 The reduction to these cases is achieved as follows. It is shown in [1] that any symmetric diameter two graph has a normal quotient graph r which is G-symmetric for some group G and which satisfies one of the following: (I) the graph r has at least one nontrivial G-normal quotient, and all nontrivial G-normal quotients of r are complete graphs (that is, every pair of distinct vertices are adjacent); or (II) all G-normal quotients of r are trivial graphs (that is, consisting of a single vertex). The context of our investigation is the following. It was shown that those that satisfy (II) fall into eight types according to the action of G [7]. One of these types is known as HA (see Subsection 2.1). In this case, the vertex set is a finite-dimensional vector space V = Fd over a prime field Fp and G = V x G0, where V is identified with the group of translations on itself and G0 is an irreducible subgroup of GL (d,p) which is intransitive on the set of nonzero vectors of V. The irreducible subgroups of GL (d,p) can be divided into eight classes Cj, i e {2,..., 9}, most of which can be described as preserving certain geometric configurations on V, such as direct sums or tensor decompositions [3]. Note that, if a diameter two graph r is G-symmetric, then the stabiliser Gv of a vertex v is not transitive on the remaining vertices since Gv leaves invariant the sets of vertices at distance 1, and distance 2, from v. Thus, in our situation, the group G0 is intransitive on the set V#, where V# := V \ {0}, the set of nonzero vectors. In paper [2] we considered the graphs corresponding to the groups G0 which are maximal in their respective classes Cj, for i < 8, and which are intransitive on nonzero vectors. (We did not consider the last class 140 Ars Math. Contemp. 13 (2017) 107-123 C9 since the groups in this class do not have a uniform geometric description.) Several classes were not considered because the maximal groups in these classes are transitive on V#, namely, the maximal groups are (a) symplectic groups preserving a nondegenerate alternating bilinear form on V, and (b) "extension field groups" preserving a structure on V of an n-dimensional vector space over Fq, where qn = pd. The aim of this paper is to examine the cases not treated in [2], namely, G0 preserves either an alternating form or an extension field structure on V, and: (III) The group G0 is irreducible and is maximal in GL (d,p) with respect to being intransitive on nonzero vectors. All quasiprimitive groups of type HA are primitive; the condition of irreducibility of G0 is necessary to guarantee that G0 is maximal in G, and hence that G is primitive. In particular, since G0 is intransitive on V#, G0 does not contain SL (V) or Sp (V). The classification in [3] can be applied to the two groups rL(n, q) and GSp (d,p): the irreducible subgroups of rL(n, q) and of GSp (d,p) which do not contain SL (n, q) and Sp (d,p), respectively, are again organised into classes C2 to C9. Again we do not consider the C9-subgroups. Observe that of the maximal subgroups of rL(n, q) in classes C2 to C8, the only transitive ones are the C3-subgroups rL(m, qn/m) with n/m prime, and the C8-subgroup rSp(n, q) of symplectic semisimilarities. We avoid these possibilities by choosing q maximal such that qn = pd. We then consider the two cases: (1) where G0 < rL(n, q) and G0 does not preserve an alternating form on F^ and (2) where G0 < TSp(n, q). Note that in this case it is possible for d/n to be not prime, and it follows from the maximality of q that G0 is not contained in a proper C3-subgroup of rL(n, q) or TSp(n, q), respectively. Since G0 is irreducible and we are not considering C9-subgroups, we now have G0 a maximal intransitive subgroup in the C (for rL(n, q) or TSp(n, q)) for some i G {2,4, 5,6,7,8}. All such subgroups of rL(n, q) for which n = d and i = 5 are considered in [2]; moreover, for some of these cases, the arguments were given in the general setting of Cj-subgroups of rL(n, q), and so can be applied here. The cases requiring the most detailed arguments are those for subfield groups and, to a lesser extent, normalisers of symplectic-type r-groups (Crgroups with i G {5, 6}). As in [2], for each family of groups G0 we have two main tasks: (i) to determine the G0-orbits, and (ii) to identify which of these orbits correspond to diameter two Cayley graphs. In the instances where we are not able to achieve either of these, we obtain bounds on certain parameters to reduce the number of unresolved cases. The rest of this paper is organised as follows: In Section 2 we give the relevant background on affine quasiprimitive permutation groups, semilinear transformations and semi-similarities. In Subsection 2.3 we present Aschbacher's classification of the subgroups of rL(n, q) and TSp(n, q). Section 3 is devoted to the proof of Theorem 1.1, which we do by considering separately the maximal intransitive subgroups in each of the classes Cj, where i G {2,4, 5, 6, 7, 8}. Notation. If A is a vector space, a finite field, or a group, A# denotes the set of nonzero vectors, nonzero field elements, or non-identity group elements, respectively. The finite field of order q is denoted by Fq. The notation used for the classical groups, some of which is nonstandard, is presented in Section 2. If r is a graph, V(r) and E(r) are, respectively, its vertex set and edge set. C. Amarra et al.: Affine primitive symmetric graphs of diameter two 141 2 Preliminaries 2.1 Cayley graphs and HA-type groups The action of a group G on a set Q is said to be quasiprimitive of type HA if G has a unique minimal normal subgroup N and N is elementary abelian and acts regularly on Q. The group G is then a subgroup of the holomorph N.Aut (N) of N (hence the abbreviation HA, for holomorph of an abelian group). It follows from [4, Lemma 16.3] that a graph r that admits G as a subgroup of automorphisms is isomorphic to a Cayley graph on N, that is, a graph with vertex set N and edge set {{x, y} | x — y G S} for some subset S of N# with S = -S and 0 G S. (Since N is abelian we use additive notation, and in particular denote the identity by 0 and call it zero.) Such a graph is denoted by Cay(N, S). If, in addition, r is G-symmetric, then S must be an orbit of the point stabiliser G0 of zero. Thus, in order for r to have diameter two, the group G0 must be intransitive on the set of nonzero elements in N. The result that is most relevant to our investigation is Lemma 2.1, which follows from the basic properties of Cayley graphs and quasiprimitive groups of type HA. Lemma 2.1 ([7]). Let r be a graph and G < Aut (r), where G acts quasiprimitively on V(r) and is of type HA. Then G = Fp x G0 < AGL (d,p) and r = Cay(F^S) for some finite field Fp, where the vector space F^ is identified with its translation group and G0 < GL (d,p) is irreducible. Moreover, r is G-symmetric with diameter 2 if and only if S is a G0 -orbit of nonzero vectors satisfying —S = S, S C V and S U (S + S) = V. The condition —S = S implies that |S + S| < |S|(|S| — 1) + 1,andif S is a Go-orbit then clearly |S| < |G0|. It follows from Lemma 2.1 that if Cay(V, S) is G-symmetric with diameter two then |V |<|S|2 + 1 <|Go|2 + 1. (2.1) This fact will be frequently used in obtaining bounds for certain parameters. In our situation pd = qn and G0 preserves on V the structure of an Fq-space; we therefore regard V as V = F^ and G0 as a subgroup of rL(n, q). 2.2 Semilinear transformations and semisimilarities Throughout this subsection assume that q is an arbitrary prime power, V is a vector space with finite dimension n over Fq, and B := {vi,..., vn} is a fixed Fq-basis of V. The general semilinear group rL(n, q) consists of all invertible maps h : V ^ V for which there exists a(h) G Fq, which depends only on h, satisfying (Am + v)h = Aa(h)uh + vh for all A G Fq and u, v G V. (2.2) The group rL(n, q) is isomorphic to a semidirect product GL (n, q) x Aut (Fq) with the following action on V: / n \9a n I £ AiVA := £ Afvf for all g G GL (n, q), a G Aut (Fq), and (2.3) \i=i J i= 1 Ai,...,An G Fq. If V is endowed with a left-linear or quadratic form then the elements of rL(n, q) that preserve ^ up to a nonzero scalar factor or an Fq-automorphism are called semisimilarities 142 Ars Math. Contemp. 13 (2017) 107-123 of That is, h is a semisimilarity of ^ if and only if for some A(h) G F# and some a'(h) G Aut (Fq), both of which depend only on h, ^(uh, vh) = A(h)^(u, v)a'(h) for all u, v G V if ^ is left-linear, and ^(vh) = A(h)^(v)a'(h) for all v G V if ^ is quadratic. It can be shown that a'(h) is the element a(h) in (2.2). The set of all semisimilarities of ^ is a subgroup of rL(n, q) and is denoted by H (n, q), where I is Sp, U, O, O+, or O-, if ^ is symplectic (i.e., nondegenerate alternating bilinear), unitary (i.e., nondegenerate conjugate-symmetric sesquilinear), quadratic in odd dimension, quadratic of plus type, or quadratic of minus type, respectively. The map a : rI (n, q) ^ Aut (Fq) defined by h ^ a(h) is a group homomorphism whose kernel GI(n, q) consists of all g G GL (n, q) that preserve ^ up to a nonzero scalar factor. The elements of GI(n, q) are called similarities of Likewise, the map g ^ A(g) for any g G rI (n, q) defines a homomorphism A from GI(n, q) to the multiplicative group F#. The kernel I( n, q) of A consists of all ^-preserving elements in GL (n, q), which are called the isometries of It should be emphasised that our notation for the similarity and isometry groups is non-standard, but follows for example [5]: the symbol GI(n, q) is sometimes used to denote the isometry group, whereas in the present paper this refers to the similarity group. In Subsection 3.1 we determine the orbits in V# of the groups rI (n, q). The following result, which gives the orbits of the isometry groups I(n, q), is useful: Theorem 2.2 ([8, Propositions 3.11, 5.12, 6.8 and 7.10]). Let V = and ^ a symplectic, unitary, or nondegenerate quadratic form on V. Then the orbits in V# of the isometry group of (V, are the sets S\ for each A G Im (^), where and SA := {v G V* | ¿(v) = A} (2.4) (¿(v,v) if $ is symplectic or unitary; (25) I ¿(v) if $ is quadratic. Observe that if $ is symplectic then ¿(v, v) = 0 for all nonzero vectors v, so it follows from Theorem 2.2 that Sp (n, q) is transitive on V*. 2.2.1 Some geometry Let f be a left-linear form on V. A nonzero vector v is called isotropic if f (v, v) = 0; otherwise, it is anisotropic. If f is symplectic or unitary, then an isotropic vector is also called singular. If f is symmetric bilinear and Q is a quadratic form which polarises to f (that is, f (u, v) = Q(u + v) - Q(u) - Q(v)), then a singular vector is a nonzero vector v with Q(v) = 0. Hence, in general, all isotropic vectors are singular and vice versa, unless V is orthogonal and q is even; in this case all nonzero vectors are isotropic but not all are singular. A subspace U of V is totally isotropic if f = 0, and totally singular if all its nonzero vectors are singular. On the other hand, a subspace U is anisotropic if all of its nonzero vectors are anisotropic. C. Amarra et al.: Affine primitive symmetric graphs of diameter two 143 For any subspace U of V we define the subspace UL := {v € V | f (u, v) = 0 V u € U} and we write V = U ± W if V = U © W and W < U^. Clearly a nonzero vector v is isotropic if and only if v € (v)^, and the subspace U is totally isotropic if and only if U < UL. A symplectic or unitary form f, or a quadratic form with associated bilinear form f, is nondegenerate (or nonsingular) if the radical VL of f is the zero subspace. A hyperbolic pair in V is a pair {x, y} of singular vectors such that f (x, y) = 1. The space V can be decomposed into an orthogonal direct sum of an anisotropic subspace and subspaces spanned by hyperbolic pairs, as stated in the following fundamental result on the geometry of formed spaces. Theorem 2.3 ([6, Propositions 2.3.2, 2.4.1, 2.5.3]). Let V = F", and let f be a left-linear form on V which is symplectic, unitary, or a symmetric bilinear form associated with a nondegenerate quadratic form Q. Then V = (xi,yi)± ... ±( xm, ym) ^ U where {xj, yi} is a hyperbolic pair for each i and U is an anisotropic subspace. Moreover: 1. If f is symplectic then U = 0. Hence n is even and, up to equivalence, there is a unique symplectic geometry in dimension n over Fq. 2. If f is unitary then U = 0 if n is even and dim (U) = 1 if n is odd. Hence up to equivalence, there is a unique unitary geometry in dimension n over Fq. 3. If f is symmetric bilinear with quadratic form Q and n is odd, then q is odd, dim (U) = 1, and there are two isometry classes of quadratic forms in dimension n over Fq, one a non-square multiple of the other. Hence all orthogonal geometries in dimension n over Fq are similar. 4. If f is symmetric bilinear with quadratic form Q and n is even, then U = 0 or dim (U) = 2. For each n there are exactly two isometry classes of orthogonal geometries over Fq, which are distinguished by dim (U). In Theorem 2.3 (4), the quadratic form Q and the corresponding geometry is said to be of plus type if U = 0, and of minus type if dim (U) = 2. 2.2.2 Tensor products Some of the subgroups listed in Aschbacher's classification arise as tensor products of classical groups. In order to describe the group action we define first the tensor product of forms. If V = U ( W, and if and $w are both bilinear or both unitary forms on U and W, respectively, then the form ( $w on V is defined by (^u ( ^w) (u ( w, u' ( w') := (u, u')^w(w, w') for all u ( w and u' ( w' in a tensor product basis of V, extended bilinearly if and are bilinear, and sesquilinearly if and are sesquilinear. If and are both bilinear then so is ( ; moreover, ( is alternating if at least one of and 144 Ars Math. Contemp. 13 (2017) 107-123 Table 2.2.4: Tensor products of classical groups I(w,¿w ) I(U ( W,^u ( ^w) Sp Oe \ Sp if the characteristic is odd; |o+ else Sp Sp O+ | O+ if ej = + for some i, or e» = — for both i; Oei < O if dim (U) and dim (W) are odd; [O- else U U U is alternating, and ^u ( is symmetric if both and are symmetric. If ^u and are both unitary then ( is unitary. The tensor product I(U, ) ( I(W, ) acts on V with the usual tensor product action — that is, for any g e I(U, ), h e I(W, ), u e U and w e W, (u ( w)(g,h) := ug ( wh. The types of forms that arise according to the various possibilities for and , which are given in terms of the possible inclusions I(U, )(I(W, ) < I(V, ), are summarised in Table 2.2.4. The tensor product of an arbitrary number of formed spaces can be defined similarly: If V = Ui (•••( Ut and ^ is a nondegenerate form on U for each i, and either all ^ are bilinear or all are sesquilinear, the form ( • • • ( is given by t (®i=1&) (®t=iUi, ®t=iwi) = JJ ^(uj,Wj) i=1 as (t=1ui and (t=1wi vary over a tensor product basis of V, extended bilinearly if the ^ are bilinear, and sesquilinearly if they are sesquilinear. Then (t=1^i is a nondegenerate bilinear (respectively, sesquilinear) form on V. If the spaces (U», are all isometric, then we can extend the results of Table 2.2.4 to the following (see [6, 9]): t o / \ |Sp (mt, q) (t=1Sp (m,q) t \ IO+ (mt, q) {O (mt, q) O- (mt,q) O+ (mt,q) (t=1U (m, q) < U (mt, q) 2.3 Aschbacher's classification if qt odd; if qt is even if qm is odd; if e = - and t is odd; else The irreducible subgroups of semisimilarity and semilinear groups are classified by Aschbacher's Theorem [3]. In [6], Aschbacher's Theorem is used to identify those irreducible subgroups which are maximal. We present below the versions that correspond to rL(n, q) C. Amarra et al.: Affine primitive symmetric graphs of diameter two 145 and to rSp(n, q). Recall that G0 does not contain either of the transitive groups SL (n, q) or Sp (n, q). Theorem 2.4. If M is a maximal irreducible subgroup of rL(n, q) that does not contain SL (n, q), then M is one of the following groups: (C2) (GL (m, q) I Sym (t)) x Aut (Fq), where mt = n; (C3) rL(m, qr), where r is prime and mr = n; (C4) (GL (k, q) g GL (m, q)) x Aut (Fq), where km = n and k = m, and the action of t is defined with respect to a tensor product basis of F^ g F^; (C5) (GL (n,q1/r) o Zq-i) x Aut (Fq), where n > 2, q is an rth power and r is prime; (C6) ((Zq_i o R).T) x Aut (Fq), where n = rf' with r prime, q is the smallest power ofp such that q = 1 (mod r), and R and T are as given in Table 2.3.5 with R of type 1 or 2; (C7) (GL (m, q) I® Sym (t)) x Aut (Fq), where m1 = n, t > 2, and the action of t is defined with respect to a tensor product basis of g^F^; (Cg) TO (n, q) or rO±(n, q) with q odd, TSp(n, q), or rU (n, q); (C9) the preimage of an almost simple group H < PrL (n, q) satisfying the following conditions: (a) T < H < Aut (T) for some nonabelian simple group T (i.e., H is almost simple). (b) The preimage of T in GL (n, q) is absolutely irreducible and cannot be realised over a proper subfield of Fq. In Theorem 2.5 the symbol [o] denotes a group of order o. In case (C2) the group [q -1] is generated by the map 5M : Xi ^ ^Xi, yi ^ yi for all xi and all yi, i e {1,..., n/2}, where ^ is a generator of the multiplicative group F# and {xi,..., xn/2,yi,..., y„/2} is a basis of F£, satisfying ^(xi, Xj) = $(yi, yj) = ^(xi, yj) = 0 whenever i = j and ^(xi, yi) = 1 for all i. Such a basis is called a symplectic basis. Theorem 2.5. If M is a maximal irreducible subgroup of rSp(n, q), then M is one of the following groups: (C2) ((Sp (m,q)4 .[q - 1].Sym (t))) x Aut (Fq), where m = n/t; or (GL (m, q) .[2]) x Aut (Fq), where m = n/2; (C3) (Sp (m, qr) .[q — 1]) x Aut (Fq), where r is prime and m = n/r; or rU (m, q2), where m = n/2 and q is odd; (C4) (GSp (k, q) x GOe(m, q)) x Aut (Fq), where q is odd, k = m, m > 3, and GOe can be any of GO, GO+, or GO_; 146 Ars Math. Contemp. 13 (2017) 107-123 (C5) (GSp (n, q1/r) o x Aut (Fq) (C6) (Zq-1 o R) .O- (2t, 2), where q > 3 and is prime, and R is of type 4 in Table 2.3.5; (C7) (GSp (m, q) Sym (t)) x Aut (Fq), where qt is odd; (C8) rO±(n, q), where q is even; (Cg) the preimage of an almost simple group H < PrL (n, q) satisfying the following conditions: (a) T < H < Aut (T) for some nonabelian simple group T (i.e., H is almost simple). (b) The preimage of T in GL (n, q) is symplectic, absolutely irreducible, and cannot be realised over a proper subfield of Fq. Table 2.3.5: C6-subgroups R T Type 1 odd Ro o ■ ■ ■ o Ro, Ro := r++2 Sp (21, r) t Type 2 2 Z4 o Qs o ■ ■ ■ o Q8 Sp (21, 2) Type 4 2 D8 o ■ ■ ■ o D8 oQ8 O- (21, 2) r t1 3 Symmetric diameter two graphs from maximal subgroups of groups rL(n, q) and rSp(n, q) In this section we prove Theorem 1.1. In view of the observations in Section 1, assume that the following hypothesis holds: Hypothesis 3.1. Let V = Fp with p prime and d > 2, which is viewed as F^ with q = pd/n for some divisor n of d (possibly d/n composite or n = d). Let H be one of the subgroups below of GL ( d, p) : 1. H = rL(n, q) = GL (n, q) x (r}, the general semilinear group on V, or 2. H = rSp(n, q) = GSp (n, q) x (t}, the group of symplectic semisimilarities of a symplectic form on V, Let t denote the Frobenius automorphism of Fq and B be a fixed Fq-basis of V, with t acting on V as in (2.3) with respect to B (with g =1 and a = t); for the case where H = rSp(n, q) assume that B is a symplectic basis of V. Define G = V x G0 < V x H< AGL (d, p) and L = G0 n GL (n, q), where G0 is a maximal Cj-subgroup of H for some i e {2,4,5,6, 7, 8} and G0 does not contain Sp (n, q) or SL (n, q). C. Amarra et al.: Affine primitive symmetric graphs of diameter two 147 We note that the groups considered in [2] are the same as the subgroups L, as defined above, of H = rL(n, q). All irreducible subgroups of GL (d, p) which are maximal with respect to being intransitive on V# thus occur as subcases of the groups considered in Hypothesis 3.1 or belong to class C9. (Indeed, G0 is maximal intransitive if n = d or if d/n is prime.) For each Aschbacher class assume that G0 = M is of the form given in Theorem 2.4 or 2.5. Since some of the other subgroups of TSp(n, q) involve classical groups, we begin with class C8. 3.1 Class Cs In this case the space V has a form 0, which is symplectic, unitary, or nondegenerate quadratic if H = rL(n, q), and is nondegenerate quadratic if H = rSp(n, q) with q even. Since the symplectic group is transitive on V#, we consider only the unitary and orthogonal cases. Throughout this section we shall use the following notation: for 0 e {□, K, #} let Se := U Sa (3.1) AeF® where the Sa are as in (2.4). If q is a square (as in the unitary case), let q0 := Jq and let Fqo denote the subfield of Fq of index 2. Also let Tr : Fq ^ Fqo denote the trace map, that is, Tr(a) = a + aqo for all a e Fq. We begin by describing the orbits of the similarity groups GI(n, q), where I e {U, O, O+, O-}. Proposition 3.1. Let V = F^, 0 be a unitary or nondegenerate quadratic form on V, and G0 = GI(n, q) with I e {U, O, O+, O-}, according to the type of 0. Let S0 be as in (2.4) and Sa, S^ and S# be as in (3.1). 1. If 0 is unitary, then the G0-orbits in V # are S0 and S#. 2. If 0 is nondegenerate quadratic, then the Go-orbits in V # are as follows: (i) S# if n = 1; (ii) S0 and S# if n is even; (iii) S0, £□ and if n is odd and n > 3. Proof. Statement 2 is precisely [2, Proposition 3.9], so we only need to prove statement 1. Assume that 0 is unitary; hence q is a square and q0 = Jq. It follows from Theorem 2.2 that S0 is a G0-orbit (that is, provided that S0 = 0), so we only need to show that S# is a G0-orbit. Let v e S#; clearly, vGo Ç S#. For any u e S# set a := f (u, u)f (v, v)-1. Then a e F#, so a = ¡qo+1 for some 3 e Fq. Hence f (u,u) = 3qo+1f (v, v) = f (¡v,3v), so by Theorem 2.2 we have u = (¡v)9 for some g e U (n, q). Then u = v^9, where ¡g e GU (n, q). Therefore vGo = S#, which proves statement 1. □ The orbits of the semisimilarity groups can be easily deduced from Proposition 3.1. Proposition 3.2. Let V = F^, 0 be a unitary or nondegenerate quadratic form on V, and G0 = rI(n, q) with I e {U, O, O+, O-}, according to the type of 0. Then for all cases, the G0-orbits are exactly the same as the GI(n, q)-orbits. 148 Ars Math. Contemp. 13 (2017) 107-123 Proof. This follows from Proposition 3.1 and the fact that the elements of ri(n, q) preserve the form up to an automorphism of Fq. □ Hence, a direct consequence of Proposition 3.2 and [2, Proposition 3.12] is: Proposition 3.3. Let r be a graph and G < Aut (r) such that G satisfies Hypothesis 3.1 with Go = rO (n, q) or G0 = rOe(n, q) (e = ±). Then r is G-symmetric with diameter 2 if and only if r = Cay(V, S) with V = F^ and the conditions listed in one of the lines 8-11 of Table 1.0.1 or lines 4-6 of Table 1.0.2 hold. We now consider the unitary case. Note that Theorem 2.3 implies that the space V contains a hyperbolic pair, which implies that there is some v e V which is nonsingular. The following are two easy but useful results which are analogous to Lemma 3.13 and Corollary 3.14 in [2]. Lemma 3.4. Let V = F^, ^ a unitary form on V, and ^ as in (2.5). Then Im = Fq0, the subfield of index 2 in Fq. Proof. Recall that f (v, v)^. = f (v, v) for any v e V, so Im < Fq0. By the preceding remarks V contains a nonsingular vector, say u. So f (au, au) = a^+1f (u, u) = n(a)f (u, u) for any a e Fq, where n : Fq ^ Fq0 is the norm map. Since n is surjective so is and the result follows. □ If ^(v, v) = 0, then (v}^ is nondegenerate and V = (v) ± (v)^. On the other hand, if ^(v, v) = 0 then (v) < (v}^. By the remarks in [6, pp. 17-18], the form ^ induces a nondegenerate unitary form on the space U := (v)^/(v), defined by (x + (v), y + (v)) := ^>(x, y) for all x, y e (v)±. It follows from [6, Propositions 2.1.6 and 2.4.1] that all maximal totally isotropic subspaces of V have the same dimension, which, in all cases, is at most n/2, so in particular v^ contains a nonsingular vector whenever n > 3. Corollary 3.5. Let V = F^, ^ a unitary form on V, ^ as in (2.5), and v e V#. Then Im ) = Fq0 if v is nonsingular and n > 2, or if v is singular and n > 3. Proof. This follows immediately from Lemma 3.4 applied to (v)±, and the remarks above. □ Proposition 3.6. Let r be a graph and G < Aut (r) such that G satisfies Hypothesis 3.1 with G0 = ru (n, q). Then r is G-symmetric with diameter 2 if and only if n > 2 and r = Cay(V, S), where V = F^ and S e {S0, S#}, with S0 and S# as in (2.4) and (3.1), respectively. Proof. By Lemma 2.1 and Proposition 3.1 we only need to prove that Cay(V, S) has diameter 2 if and only if n > 2. If n =1 then V is anisotropic, so GU (n, q) is transitive on V# by Proposition 3.1 (1) and Cay(V, S) is a complete graph. If n > 2 then V# \ S0 = S# and V# \ S# = S0 by Proposition 3.1. Claim 1: S# C S0 + S0. Let v e S#. Then by Corollary 3.5 there exists u e (v) with ^>(u) = —^>(v). Set w := ^(u + v), where ft := a^(v)-1 and a e Fq such that Tr(a) = ^(v). Then w, v — w e S0, so v e S0 + S0 and therefore S# C S0 + S0. Claim 2: S0 C SM + SM for any ^ e (Im (^))#. Let v e S0. Suppose first that n > 3. _L Then by Corollary 3.5, for any ^ e (Im (^))# there exists w e SM n (v)±. It is easy to _ _ )X. verify that ^(v — w) = ^(w), so v — w e SM and v e SM + SM. Therefore S0 C SM + SM. C. Amarra et al.: Affine primitive symmetric graphs of diameter two 149 If n = 2 then (v)^ = (v) for any v e S0. We show that there exists u e S0 such that ^(u, v) = 1. Indeed, take x e V \ (v). Then ^(v, x) = 0. If x e S0 define u' := x; if x G S0 let u' := av + ^(v, x)x where a e Fq with Tr(a) = —^(x). Then in both cases u' e S0 and ^(u', v) = 0, and we take u to be the suitable scalar multiple of u' such that ^(u,v) = 1. Let w := ^u + yv, where ^,7 e Fq with Tr(^) = 0 and Tr(^qo7) = Then w, v — w e SM, and thus v e SM + SM. Therefore S0 C SM + SM. It follows from Claims 1 and 2, respectively, that Cay(V, S0) and Cay(V, S#) both have diameter 2. This completes the proof. □ 3.2 Class C2 In this case V = ©f=1Uj, where Uj = F^ for each i, mt = n and t > 2. Assume that B = U t =1 Bj, where Bj is a basis for Uj for each i. We write the elements of V as t-tuples over F^; under this identification the t-action is equivalent to the natural componentwise action. Assume first that H = rL(n, q). It turns out that the G0-orbits in V# are the same as the L-orbits, and thus the graphs that we obtain are precisely those in [2, Proposition 3.2]. Lemma 3.7. Let G0 be as in case (C2) of Theorem 2.4. Then the G0-orbits in V # are the sets Xs for each s e {1,..., t}, where Xs := {(u1,..., ut) e V# | exactly s coordinates nonzero}. (3.2) Proof. Let v e Xs. Clearly vGo C Xs; since vL = Xs by [2, Lemma 3.1] it follows that vGo = Xs. □ Proposition 3.8. Let r be a graph and G < Aut (r) such that G satisfies Hypothesis 3.1, with H = rL(n, q) and G0 as in case (C2) of Theorem 2.4. Then r is G-symmetric with diameter 2 if and only if r = Cay(V, Xs), where Xs is as in (3.2), such that qm > 2 and s > t/2. Proof. This follows immediately from Lemma 3.7 and [2, Proposition 3.2]. □ We now consider the case where H = TSp(n, q) with n > 4. By Theorem 2.5 there are two types of C2-subgroups, corresponding to two kinds of decompositions. We refer to these subcases as (C2.1) and (C2.2). (C2.1) The dimension m of the subspaces Uj is even, Uj is a symplectic space for each i, the subspaces Uj are pairwise orthogonal, and Go = {(gi,... ,gt)na | n e Sym (t), a e (t), gj e GSp (m, q), A(gj) = A(gi)} = (Sp (m,q)4 .[q — 1].Sym (t)) x (t), (3.3) where A : GSp (n, q) ^ F# is as defined in Subsection 2.2. (C2.2) The dimension m = n/2 so that t = 2, both subspaces Uj are totally singular with dimension n/2, q is odd if n = 4, and Go = {(g,g-T) na | n e Sym (t), a e (t), g e GL (m, q^ (34) = (GL (m,q) .[2]) x (t), (.) where gT denotes the transpose of g, and g-T = (gT)-1. 150 Ars Math. Contemp. 13 (2017) 107-123 Lemma 3.9. For each s € {1,..., t} let Xs be as in (3.2). The G0-orbits in V # are 1. the sets Xs for each s € {1,..., t} if case (C2.1) holds and G0 is as in (3.3); 2. the sets Xi and |J CTe/T\ for all ß € Fq, if case (C2.2) holds and G0 is as in (3.4), where Wß := (wi,xß)L (3.5) L = Go n GL (n,q) = GL (m, q) .[2], wi := (1,0,..., 0) G Fm and x^ G (F^)# with first component ft. Proof. The proof of part (1) is similar to that of [2, Lemma 3.1] and uses the transitivity of Sp (m, q) on U#, so we only need to prove part (2). Assume that case (C2.2) holds. Then L = K.Sym (2), where K := {(g, g-T) | g G GL (m, q)}. It is easy to see that Ui © {0} and {0} © U2 are K-orbits, so Xi = (Ui {0}) U ({0} © U2) is a Go-orbit. Let (u, v) G X2, and for any ft G Fq define wß := (ß, 0,..., 0) if ß = 0, (0,1, 0,..., 0) if ß = 0. (3.6) Since w1 G wGL(m'q) we can assume that u = w1. Suppose that v = (ft, v2,..., vm). Claim 1: (w1, y) G (w1, v)K if and only if y = (ft, y2,..., ym) for some y2,..., ym G Fq. Indeed, (w1, y) G (w1, v)K if and only if y = vh for some h G StabGL(m,q)(w1). Now wh = w1 if and only if the matrix of h-T has the form 1 C \ 0 D 0 ) where C is a 1 x (m — 1) matrix over Fq and D € GL (m — 1, q). Clearly, the orbit of v under the subgroup {h-T | h € StabGL(m,q) (wi)} is the set of all nonzero vectors in F^ with first component ß. Therefore Claim 1 holds. Claim 2: (wi, v)L = (wi, v)K. By Claim 1 we can assume that v = wß. If ß = 0 let ß g := 0 0 Im— i If ß = 0 let g := (0 0) if m = 2, and 0 1 g := \ / if m > 2. Then g G GL (m, q) for all cases, and wg = wg = v. Hence (wg, vg ) = (v,w1), so that (v,w1) G (w1,v)K. Therefore (w1,v)L = (w1,v)K U (v,w1 )K = (w1, v)K, which proves Claim 2. 0 0 0 Im-2 C. Amarra et al.: Affine primitive symmetric graphs of diameter two 151 It follows from Claims 1 and 2 that each set Wp is an L-orbit (and moreover Wp = Wp> if and only if p = p'). It follows from the definition of the t-action on V# that (wi,v)Go =U £ ■e W«°- . This completes the proof of part (2). □ Proposition 3.10. Let r be a graph and G < Aut (r) such that G satisfies Hypothesis 3.1 with H = rSp(n, q) and i = 2. Then r is G-symmetric with diameter 2 if and only if r = Cay(V, S), where 1. if case (C2.1) holds, then qm > 2, G0 is as in (3.3), S = Xs, and s > t/2; 2. if case (C2.2) holds with qm = 2, then G0 is as in (3.4), and S = Wp for any p G Fq ; 3. if case (C2.2) holds with qm > 2, then G0 is as in (3.4), and S = Xi or S = Uae(r> Wp- for some p G Fg; with Xs as in (3.2) and Wp as (3.5). Proof. The graph of (1) is precisely that of Proposition 3.8, and the fact that it is G-symmetric follows from Lemma 2.1. So assume that case (C2.2) holds. By Lemma 2.1 we only need to show that V = S U (S + S) unless S = X1 and q = 2. It follows from Proposition 3.8 (with t = 2) that Cay(V, X1) has diameter 2 (with G quasiprimitive) if and only if qm > 2, which proves part of statement (3). Thus we may assume that S = Uae(j> Wp- for some p G Fq. It remains to prove that V = S U (S + S). Let wp be as in (3.6) and 7 G Fq, with 7 = p. Define go := 1 1 0 1 and h0 := 0 -1 -1 Yo where y0 := 1 - ft 1y if P = 0 and yo := 0 if ^ = 0. If m = 2 let g := g0 and h := h0; if m > 3 define g and h by 'go 0 0 Im-2 , g : = and / h := ho \ 0 1 'm-2 / Then g, h G GL (m, q) for all m > 2, and wg + wh = w1. Recall that q is odd if m = 2 so we can take x g (F^ )# where wp if p = 0; (0, —y/2) if p = 0 and m = 2; (0,0,1,0,..., 0) if p = 0 and m > 3. Then for all cases y := xg + xh has first component y. Hence, applying Lemma 3.9, we have W7 = (wi, y)L C Wp + Wp for any y = p. Since also {0} U Xi C Wp + Wp, it follows that V = Wp U (Wp + Wp). Therefore V = S U (S + S), which completes the proof of parts (2) and (3). □ 0 152 Ars Math. Contemp. 13 (2017) 107-123 3.3 Class C4 In this case V = U < W = F^ < F^ with k, m > 2, and B is a tensor product basis of V, that is, B = {wj < wj | 1 < i < k, 1 < j < m}, where Bu := {ui,..., uk} and Bw := {wi,..., wm} are fixed bases of U and W, respectively. We choose t to fix each of the vectors uj < wj. Then for any simple vector u < w G V, we have (u < w)T = uT < wT, and for any v = J2¿=1 (a < G V, r vT = £ < < bT. j=i Recall that k = m in the description given in Theorems 2.4 and 2.5; however, all of the results in this section also hold for k = m, so we do not assume that k and m are distinct. In this way the results yield useful information for the C7 case. A nonzero vector in V is said to be simple in the decomposition U < W if it can be written as u < w for some u G U and w G W. The tensor weight wt(v) of v G V#, with respect to this decomposition, is the least number s such that v can be written as the sum of s simple vectors in U < W. It follows from [2, Lemma 3.3] that wt(v) < min {k, m} for any v G V#, and that for each s G {1,..., min {k, m}} there is a vector v G V# with weight s. The proof of the following observation is straightforward and is omitted. Lemma 3.11. For any v G V# and any a G Aut (Fq), wt(vCT ) = wt(v). Assume first that H = rL(n, q). As in the previous section, the G0-orbits in V# are the same as the L-orbits. This follows easily from Lemma 3.11 and the results in [2]. Lemma 3.12. Let G0 be as in case (C4) of Theorem 2.4. Then the G0-orbits in V# are the sets Ys for each s G {1,..., min {k, m}}, where Ys := {v G V#|wt(v) = s} . (3.7) Proof. This is a consequence of Lemma 3.11 above, and of [2, Lemmas 3.3 and 3.4]. □ We then obtain the same graphs as those in [2, Proposition 3.5]. Proposition 3.13. Let r be a graph and G < Aut (r) such that G satisfies Hypothesis 3.1 with G0 as in case (C4) of Theorem 2.4, where k and m may be equal. Then r is G-symmetric with diameter 2 if and only if r = Cay(V, Ys), where s > ^ min {k, m} and Ys is as in (3.7). Proof. This follows immediately from Lemma 3.12 and [2, Propositon 3.5]. □ Now assume that H = rSp(n, q). In this case k is even, m > 3, q is odd, and ^ = ^u < ^w, where ^u is a symplectic form on U and ^w is a nondegenerate symmetric bilinear form on W. We can choose Bu and Bw appropriately so that B is a symplectic basis and hence we can again choose t to fix each of the vectors uj < wj. The G0-orbits C. Amarra et al.: Affine primitive symmetric graphs of diameter two 153 in this case are proper subsets of the sets Ys in (3.7), and are in general rather difficult to describe, as are the L-orbits. For instance, if v = J2¿=1 a ® b € Ys, it is easy to see that vGo £ aj ® bj aj € U bj € b,G°'(m'q> „ i=1 If s = 1 then the set Y1 of simple vectors splits into the G0-orbits Y/, where 0 € {0, #} if m is even and 0 € {0, □, K} if m is odd, and Yl := {a b | a € U#, b € Se} . If s > 1 suppose that exactly r of the vectors b belong in S# for some r, 0 < r < s; if m is odd suppose further that exactly rn belong in Sn and rH in SH. If m is even then vGo c Ysr, where Ysr := | £ aj ® bj € Ys exactly r of the vectors bj are in S# |, and if m is even then vGo c YTD'rH, where YsrD'r® := j£ aj ® bj € Y exactly re of the vectors bj are in Se for 0 € {□, K} j . The sets Ysr and Y/Dabove are, in general, not G0-orbits. For instance, if s = 2, the weight-2 vectors a1 ® b1 + a2 b2, al ® bl + a2 b2 € Y20 (or Y20'0 if m is even), such that b1 ± b2 and b1 / b2, belong to different G0-orbits. The following is an easy consequence of the preceding discussion. However, as discussed, we do not have a good description of the Go-orbits. Proposition 3.14. Let r be a graph and G < Aut (r) such that G satisfies Hypothesis 3.1 with G0 as in case (C4) of Theorem 2.5, where k and m may be equal. If r is G-symmetric with diameter 2, then r = Cay(V, S) where S = vGo for some v € Ys, where Ys is as in (3.7) and s > 2min {k, m}. Proof. This follows immediately from the discussion above together with Proposition 3.13. □ 3.4 Class Cb In this case n > 2, d/n is composite with a prime divisor r, and V has a fixed ordered basis B := (v!,...,v„). Let q0 := q1/r and let Fqo denote the subfield of Fq of index r. Let V0 be the Fqo-span of B. Then V0 is a vector space over Fqo that is contained in V, but V0 is not an Fq-subspace of V. To any v = J2"=1 € V we can associate the Fqo-subspace Dv of Fq, where Dv := (a1,...,a„)F,o. (3.8) 154 Ars Math. Contemp. 13 (2017) 107-123 Set c(v) := dimFq0 (Dv ), (3.9) and note that c(v) < min {r, n}. For any A G Fq it is clear that DXv = ADv, so c(Av) = c(v), and it is also easy to show that c(vCT) = c(v) for any a G Aut (Fq). Let [Dv] := {ADv | A G F# }, and observe that Du g [Dv^ ] if and only if Du = ADv^ = 1 Dv j for some A G F#. Hence D (Duy Xa Dv, so that D -i G [Dv]. Thus [D^] = [Dv]a. 3.4.1 Case H = rL(n, q) By Theorem 2.4 Go = (GL (n,qo) o Z,_i) x (r> and L = GL (n, q0) o Zq-1. Regard the field Fq as a vector space of dimension r over Fq0, and for any a G {1,..., r}, define N Fq if a = r, K(a) = { q I Fqo otherwise. For a G {1,..., r} define where n(a) := |F# : K(a)#| (3.10) (3.11) i- 1 := qo - qô <=0^3 qô the number of a-dimensional subspaces of Fro. In particular n(r) = n(1) = 1. Lemma 3.15 gives some elementary observations about K(a) and n, whose significance will be apparent in Corollary 3.19. The proof of Lemma 3.15 is straightforward and is omitted. Lemma 3.15. Let Fq0 be a proper nontrivial subfield of Fq with prime index r, and suppose that Fq is viewed as a vector space over Fq0 with dimension r. For any a G {1,..., r}, let D denote the set of all Fq0 -subspaces of Fq with dimension a, and let K(a) and n(a) be as defined in (3.10) and (3.11), respectively. Then the following hold: 1. For any D gD, {X G Fq | XD = D} = K(a). 2. For any D G D, the sets [D] = {AD | A G F#} partition D. Moreover, |[D]| = |F# : K(a)# |, and the number of distinct parts [D] in D is n(a). The main result for this case, which relies on the value of the parameter c(v), is the following. It shows that examples do exist. r a C. Amarra et al.: Affine primitive symmetric graphs of diameter two 155 Proposition 3.16. Let r be a graph and G < Aut (T) such that G satisfies Hypothesis 3.1 with H = rL(n, q) and i = 5. Then r is connected and G-symmetric if and only if r = Cay(V, vGo ) for some v G V#. Moreover, if Dv and c(v) are as in (3.8) and (3.9), respectively, then the following hold. 1. If c(v) = r or c(v) = r — 1 then diam(r) = 2. 2. If c(v) = 1 then diam(r) = min {n, r}. In particular diam(r) = 2 if and only if n = 2 or r = 2. 3. If 2 < c(v) < 2min {n, r} then diam(r) > 2. 4. Let n be as defined in (3.11), s be the largest divisor of d/n with s < n(c(v)), and ,, , J 18s/17 if qo = 2; [s — 5/4 if qo > 2. If 3 < n < r and n/2 < c(v) < (r(n — 2) + ki(qo))/(2n), then diam(r) > 2. The cases not covered by Proposition 3.16 are discussed briefly at the end of the section. The proof of Proposition 3.16 is given after Lemma 3.20, and relies on several intermediate results. We begin by describing the GL (n, q0)-orbits in terms of the subspaces Dv, which in turn leads to a description of the G0-orbits in V#. Lemma 3.17. For any v G V# let Dv and c(v) be as in (3.8) and (3.9), respectively, and let U denote the set of all Fqo-independent c(v)-tuples in V0. Then for any fixed Fqo -basis {A,... ,Pc(v)} of Dv, ( c(v) vGL(n,qo) = Aui (ui,...,uc(v)) GU = {u G V# | D„ = Dv } . Proof. Suppose that v = ^"=1 a^. Define U := {u G V#|D„ = Dv } (3.12) and ( c(v) W := PiUi (ui,...,uc(v)) GU. (3.13) Claim 1: vGL(n,qo) Ç U. Let g G GL (n, q0) with matrix [gjk] with respect to B. Then vg = J2n=i a'kvk, where ak = ^"=1 ajgjk G Dv for each k. Hence DvS < Dv. Since v and g are arbitrary, we also have Dv < DvS. So DvS = Dv, and therefore vGL(n,qo) Ç U. Claim 2: U Ç W. Let u = J2"=i ajvj G U. Writing aj = J2C:SVi) P^j for each j, where all Yij G Fw, we get u = ^Él PiUi, with Ui = £j=i Yijvj G V0 for all i. It remains to show that the set u := {ui,..., uc(v)} is Fqo-independent. Indeed, let {ui,..., ub} be a maximal Fqo -independent subset of u, and extend this to an ordered Fqo -basis B' := (ui,..., ud) of V0. Then u = J2t=i Pku'k for some Pi,..., Pb G Fq, and 156 Ars Math. Contemp. 13 (2017) 107-123 if g € GL (n, q0) is the change of basis matrix from B' to B, then ug = ^fc=1 P'kvfc- So Du = Dug by Claim 1, and thus b < c(v) = dimF (Du) = dimF (Dug) < b. Hence b = c(v) and u is Fq0 -independent. Therefore U C W. Claim 3: W C vGL(n'qo). It is easy to see that W is contained in one orbit of GL (n, q0), and it follows from Claims 1 and 2 that v G W .So W C vGL(n'qo), as claimed. Thus we have vGL(n'qo) = U = W by Claims 1-3. □ Proposition 3.18. For any v € V# let Dv and c(v) be as in (3.8) and (3.9), respectively, and let U be the set of a// Fqo-independent c(v)-tuples in V0. Then for any fixed Fqo -basis i^i,... ,&(v)} of Dv we have c(v) A E ßi i=i (ui,...,uc(v)} eM, A G F* = {u G V* | Du = ADv, A G F*} and (ui, ...,uc(v) ) G M, A G F*, a G (t ) c(v) A E ß7 i=i {u G V* I Du = A(Dv)7, A G F*,a G (t)} . (3.14) Proof. Let U' := {u € V# | Du = ADv for some A € F#}. Since L = GL (n, qo) o Z9_i and D^v = ADv for any A € Fq, it follows from Lemma 3.17 that vL = U'. Thus vGo = U {u7 7£{r > u G vL} Ç W' where W' := {u € V# | Du = A(Dv)CT, A € F#, a € (t)}. For any w € W with Dw = ^(Dv)p for ^ € F# and p € (t), we have w € (vp)L C vGo. Therefore vGo = W', and the rest follows from Lemma 3.17. □ Corollary 3.19. Let v € V#, and let K, n, Dv and c(v) be as defined in (3.10), (3.11), (3.8) and (3.9), respectively. 1. For a € {1,..., min {n, r}}, the number of orbits vL with c(v) = a is n(a). 2. |vL c(v) |GL (c(v),qo)| • |F* : K(c(v))#| 3. |vG s |vL | for some divisor s of d/n with s < n(c(v)). Proof. It follows from Proposition 3.18 that the map ^ [Dv] := {ADv | A G F#} is a one-to-one correspondence between the set of L-orbits and the set of classes [D] of Fq0-subspaces of Fq. Therefore, by Lemma 3.15 (2), there are exactly n(a) orbits with c(v) = a, which proves part (1). Also by Proposition 3.18, we have |vL| = |U| • |[Dv]|, where U is the set of Fq0 -independent c(v)-tuples in V0. So |U| n c(v) |GL (c(v),qo) |, L v G o v= n C. Amarra et al.: Affine primitive symmetric graphs of diameter two 157 and by Lemma 3.15 (2), |[D„]| = |F# : K(c(v))#|. This proves part (2). Since L < Go we must have |vGo | = s |vL| for some s dividing |G0 : L| = |Aut (Fq) | = d/n. Also s < n(c(v)) since c(vCT) = c(v), which proves part (3). □ Lemma 3.20. Let r = Cay(V, vGo) for some v e V#, and let c(v) be as in (3.9). Let w e V. 1. If w e vGo + vGo then c(w) < 2c(v). 2. If Dw < Dv then w e vGo + vGo. Proof. Let U and W denote the sets of Fqo -independent c(v) - and c(w)-tuples, respectively, in V. 0 Suppose first that w = x + y for some x, y e vGo. Then by Proposition 3.18 we can write x and y as x = J2¿Si A/j^ and y = ^¿Si y/j7'yj for some scalars A, y e F#, maps p, a e Aut (Fq), and c(v)-tuples (x1,..., xc(v)) , (y1,..., yc(v)) e U. Hence Dw = Dx + y C (a/P, ..., A/P(v), u/i,..., u/%)) , and therefore c(w) = c(x + y) < 2c(v). This proves part (1). To prove part (2), observe that Lemma 3.17 implies that we can write v and w as v = ^¿Si YiUi and w = ^¿£w) ¿¿zj for some («4,..., wc(v)) e U and (zi,..., zc(w)) e W, and for some fixed Fqo-bases {7i, ..., 7C(v)} and {¿1,..., ¿c(w)} of Dv and Dw, respectively. Since Dw < Dv then c(w) < c(v), and we can extend {¿1,..., ¿c(w)} to an Fqo -basis {¿1,..., ¿c(v)} of Dv, and (z1,..., zc(w)) to (z1,..., zc(v)) e U. Set x := ¿¿zj and y := ^¿Si ¿¿yj, where yj := zj+i - zj if 1 < i < c(w) - 1, yc(w) := zi, and yj := -zj if c(w) + 1 < i < c(v). Then (yi,... ,yc(v)) e U and Dx = Dy = Dv, so by Lemma 3.17 we have x, y e vGL(n'qo) C vGo. Therefore x + y e vGo + vGo. Now Dw = Dx+y, so applying Lemma 3.17 again we get w e (x + y)GL(n,qo) C vGo + vGo. Thus (2) holds. □ Proof of Proposition 3.16. Suppose that r-1 < c(v) < r. Observe that n(r -1) = n(r) = 1, so for either value of c(v) we have vL = {u e V | c(u) = c(v)}, which in turn implies that vGo = vL. If c(v) = r then Dv = Fq, and clearly Dw < Dv for any w e V# \ vGo. So w e vGo + vGo by part (2) of Lemma 3.20, and thus V# \ vGo C vGo + vGo. Therefore diam(r) = 2. Now suppose that c(v) = r - 1, and let w e V # \ vGo. If c(w) < r - 1 then it follows from part (1) of Corollary 3.19 that Dw < ADv = for some A e F#. Thus w e (Av)Go + (Av)Go = vGo + vGo by Lemma 3.17. If c(w) = r let x := ^¿-i ajvj and y := /¿vi + Yvr, where {ai,..., ar-i} is an Fqo-basis of Dv, 7 e F# \ Dv, and = | aj+i - a if 1 < i < r - 3; 1 ai - ar-2 if i = r - 2. Then c(x) = c(y) = r-1 and c(x+y) = r, so x, y e vGo and w e (x+y)Go C vGo +vGo. Therefore V# \ vGo C vGo + vGo, and again we have diam(r) = 2. This completes the proof of part (1). If c(v) = 1 then we get the special case vL = vGo = (FqV0)#. Let distr(0V, w) denote the distance in r between the vertices 0V and w; we claim that distr(0V, w) = 158 Ars Math. Contemp. 13 (2017) 107-123 c(w) for any w G V. Let ¿(w) := distr(0V, w). Then w G Y by Proposition 3.18, where Y is as in (3.13), so w can be written as a sum of c(w) elements of (FqV0)# and thus ¿(w) < c(w). On the other hand w = ^¿=1 A^, where Aj G F# and w G V0# for all i. Writing each uj as uj = J2"=i wj where G Fqo for all i, j, we get w = Ajwj where Aj = Aj^jj for each j. Hence Dw < (Ai,..., A£(w))f,0 , so that c(w) < ¿(w). Therefore ¿(w) = c(w), as claimed. It follows immediately that diam(r) = min {n, r}, and that diam(r) = 2 if and only if n = 2 or r = 2. This proves (2). Suppose that diam(r) = 2. Then c(w) < 2c(v) for any w G V# by part (1) of Lemma 3.20, and in particular 2c(v) > min {n, r} since there clearly exists u G V# with c(u) = min {n, r}. Hence c(v) < 1 min {n, r} implies that diam(r) > 2, and part (3) holds. Finally, let a := c(v), S := vGo, and n(a) as in (3.11). By Corollary 3.19 we have |S|< [ n 1 |GL (a,qo) ||F# : F# | s, L J q o where s is the largest divisor of d/n with s < n(a). Hence |S|2 + 1 < q2an |F# : F#0|2 s2. Observe that s < qg4 for all s > 1, where t = 17 if q0 = 2, and t = 2 if q0 > 3. Also, for q0 > 3, we have q0 - 1 > q^8, so that |F# : F# | < q,T-5/8. With these bounds we obtain |S|2 + 1 < qo2(an+r)+ki(qo) where k1(q0) is as defined in (4). It is easy to verify that if a < (r(n - 2) - k1(q0))/(2n) then 2(an + r) + k1(q0) < rn, so |S|2 + 1 < |V|, and thus diam(r) > 2 by Lemma 2.1. This proves part (4). □ Remark 3.21. Some small cases covered by Proposition 3.16 are summarised in Table 3.4.6. The cases left unresolved by Proposition 3.16 are the following: 1. 5 < r < n, r/2 < c(v) < r — 2; 2. 2 = n < r - 2, c(v) = 2; 3. 3 < n < r, maxn/2, (r(n - 2) - k1(q0))/(2n) < c(v) < r - 2. Let a := c(v) < r, S = vGo, and s as in Proposition 3.16 (4). Then s > 1, |F# : F#o|>qS-2 and n q o |GL (a,qo) | >q2a(n-1), so |2 |Go|2 + 1 > ([ n |GL (a, qo) ||F# : F#1 s) +1 > q2o(n-1) + 2(r-2) > qo . It is easy to show that if condition (1) or (2) holds then 2(a(n - 1) + r - 2) > rn, and thus |G012 + 1 > |V|. This, unfortunately, does not lead to any conclusion about diam(r). a C. Amarra et al.: Affine primitive symmetric graphs of diameter two 159 Table 3.4.6: r as in Proposition 3.16 for small values of r and n r n c(v) Conclusion about r = Cay(V, vGo ) 2 > 2 1 diam(r) = 2 by Proposition 3.16 (2) 2 diam(r) = 2 by Proposition 3.16 (1) 3 2 1 diam(r) = 2 by Proposition 3.16 (2) 2 diam(r) = 2 by Proposition 3.16 (1) 3 > 3 1 diam(r) = 3 by Proposition 3.16 (2) 2 diam(r) = 2 by Proposition 3.16 (1) 3 diam(r) = 2 by Proposition 3.16 (2) 5 2 1 diam(r) = 2 by Proposition 3.16 (2) 5 3 1 diam(r) = 3 by Proposition 3.16 (2) 5 4 1 diam(r) = 4 by Proposition 3.16 (2) 4 diam(r) = 2 by Proposition 3.16 (1) 5 > 5 1 diam(r) = 5 by Proposition 3.16 (2) 2 diam(r) > 2 by Proposition 3.16 (3) 4 diam(r) = 2 by Proposition 3.16 (1) 5 diam(r) = 2 by Proposition 3.16 (1) 3.4.2 Case H = rSp(n,q) By Theorem 2.5, Go = (GSp (n,qo) o Zq_i) x (t> and L = GSp (n, q0) o Zq-1. The main result in this section is parallel to part (4) of Proposition 3.16. Proposition 3.22. Let r be a graph and G < Aut (r) such that G satisfies Hypothesis 3.1 with H = rSp(n, q) and i = 5. Then r is connected and G-symmetric if and only if r = Cay(V, vGo) for some v e V#. Moreover, if s := |t| = |G0 : L| and c(v) is as defined in (3.9), and if t :f 9/17 if qo = 2, [1/2 if qo > 2 then the following hold: 1. If c(v) < lmin {n, r} then diam(r) > 2. 2. If 3 < n < r, c(v) > n/2 and r > (n2 + n + 2st)/(n — 2), then diam(r) > 2. Proof. Assume that c(v) < 2min {n, r}. Let S = vGo, and let r' = Cay(V, vG°), such that G' satisfies Hypothesis 3.1 with H = rL(n, q) and i = 5. Then r is a subgraph of r', and hence diam(r) > diam(r'). If c(v) = 1 then diam(r') > min {n, r} > 2 by part (2) of Proposition 3.16, and if c(v) > 2 then diam(r') > 2 by part (3) of Proposition 3.16. In both cases diam(r) > 2. This proves statement (1). We now prove statement (2). Observe that for any A e F# and g e GSp (n, q0), we have Avg = vXg e vGSp(n,qo) if and only if A1n e Zqo-1, the subgroup of scalar 160 Ars Math. Contemp. 13 (2017) 107-123 matrices in GL (n, q0). Hence vL = |JA£AvGSp(n'qo) can be written as a disjoint union vL = |JXeT AvGSp(n'qo), where T is a transversal of F#o in F#. Thus |vL| < |T||GSp (n,qo) | = (q0 - 1)|Sp (n, qo) | and |S| < s|vL|, where s = |G0 : L|. We have n/2 - „"2/4 IT 2i -i \ . Jn2+n)/2 |Sp (n, qo) | = q?^ (q2* - 1) < q0" Also, as in the proof of Proposition 3.16 (4), we have s < qg4 for any s, where t = 17 if q0 = 2, and t = 1 if q0 > 3. Hence |S|2 + 1 < s2(q° - 1)2q°2+° < q°2+°+2o+2si. If r > (n2 + n + 2st)/(n - 2) then rn > n2 + n + 2r + 2st, so |V| > |S|2 + 1 and diam(r) > 2 by Lemma 2.1. Therefore part (2) holds. □ 3.5 Class Ce In this case dim (V) = r4 where r is a prime different from p, q is the smallest power of p such that q = 1 (mod |Z(R)|) for some R in Table 2.3.5, and G0 = (Z,_i o R).T x (r>, with T as in Table 2.3.5. By Theorems 2.4 and 2.5, if H = rL(n, q) then R is of type 1 or 2, and if H = TSp(n, q) with q odd then R is of type 4. Proposition 3.23 is an extension of [2, Proposition 3.6], and is proved somewhat similarly. Proposition 3.23. Let V and G0 be as above, and let r := Cay(V, S) for some G0-orbit s c v #. 1. Suppose that r is odd, q = 1 (mod r), and R is Type 1. If diam(r) = 2 then 1 < t < 3, r < r0(t), and q < q0(r, t), where r0(t) and q0(r, t) are given in Table 3.5.7. 2. Suppose that r = 2, t > 2, q = 1 (mod 4), and R is Type 2. If diam(r) = 2 then 2 < t < 6 and q < q0(t), where q0(t) is given in Table 3.5.8. 3. Suppose that r = 2, t > 2, q is odd, and R is 7ype 4. If diam(r) = 2 then 2 < t < 7 and q < q0(t), where q0(t) is given in Table 3.5.9. 4. Suppose that r = 2, t = 1, q is odd, and R is 7ype 2 or 4. Then diam(r) = 2 for any S. Proof. If q = p£ and R is Type 1 or 2, then |G0| = ¿(q - 1)r2t|Sp (2t, r) | < ¿(q - 1)r2t2+3t. C. Amarra et al.: Affine primitive symmetric graphs of diameter two 161 Table 3.5.7: Bounds for r and q when R is Type 1 t 1 2 3 r0(t) 11 3 3 q0(3,t) 186619 73 11 q0(5,t) 521 - - q0(7,t) 71 - - q0(11,t) 23 - - Table 3.5.8: Bounds for q when R is Type 2 and t > 2 t 2 3 4 56 q0(t) 23029 569 73 17 5 Table 3.5.9: Bounds for q when R is Type 4 and t > 2 t 2 3 4 567 q0(t) 1913 149 37 11 5 3 Suppose first that R is Type 1. In this case r is odd and q = p = 1 (mod r), so I < r - 1, q > r, and |Go|2 + 1 < ((q - 1)r2i2+3i+!)2 + 1 < q4t2+6i+4. It can be shown that 4t2 + 6t + 4 < rl for the following cases: t > 5 and r > 3, t = 1 and r > 17, t = 2 and r > 7, and t G {3,4} and r > 5. Thus for all these cases |G0|2 + 1 < |V |. For all remaining pairs (r, t) define n(q,r,t) := ((r - 1)(q - 1)r2t|Sp (2t, r) |)2 + 1 - qr'. Then |Go|2 + 1-|V| < n(q,r,t) andn(q,r,t) < 0if q > ((r - 1)r2t|Sp (2t,r) |)2/(r'-2). Getting the largest prime power q = p = 1 (mod r) less than or equal to this bound, with I < r - 1 and n(q, r, t) > 0, gives the values q0(r, t) in Table 3.5.7, and for each t we take r0(t) to be the largest value of r for which there exist such q. In particular, n(q, r, t) < 0 for the following cases: (r, t) = (13,1) and q > 13, (r, t) = (5, 2) and q > 7, (r, t) = (3,4) and q > 3; for these cases there is no value of q less than or equal to the given bound that satisfies all the required conditions. This proves part (1). Now suppose that R is Type 2 with t > 2. Then r = 2 and q = p = 1 (mod 4), so I < 2, q > 4, and |Go|2 + 1 < ((q - 1)22t2+3t+1 )2 + 1 < q2t2+3t+3. We have 2t2 + 3t + 3 < 24 whenever t > 7, hence |G0|2 + 1 < |V| for all such t. For t g {1,..., 6} define n(q,t) := (2(q - 1)22t|Sp (2t, 2) |)2 + 1 - q2', 162 Ars Math. Contemp. 13 (2017) 107-123 and observe that |Go|2 + 1 - |V| < n(q,t) < 0 for all q > (22i+1|Sp (2t, 2) |)1/(2' 1 X). The values of q0(t) in Table 3.5.8 are the largest prime powers q = pe = 1 (mod 4) less than or equal to these bounds, with I < 2 and satisfying n(q, t) > 0. This proves (2). For (3), suppose that R is Type 4 with t > 2. Then r = 2 and |Z(R)| = 2, so I = 1 and q = p. Also q > 3, so q3/2 > 4. We have |Go| = (q - 1)22t |O- (2t, 2) I < (q - 1)22i2+i+2 so |Go|2 + 1 < ((q - 1)22i2+i+2)2 + 1 < q242i2+4+2 < q3t2 +3 We have 3t2 + 11 + 5 < 24 (and hence |G012 + 1 < | V|) for all t > 8. For t € {2,..., 7} define ( | |) n(q,t) := ((q - 1)22t |O- (2t, 2)|)2 + 1 - q2'. Then |Go|2 + 1 - |V| < n(q,t) < 0 for all q > (22t |O- (2t, 2)|)1/(2'-1-1). As in the previous cases we take q0(t), 2 < t < 7, to be the largest prime q less than or equal to these bounds such that n(q, t) > 0. This yields Table 3.5.9 and proves (3). Statement 4 for the case where R is type 2 is precisely [2, Proposition 3.6 (2)]. For the case where R is type 4 define the matrices a, c € GL (V) by -01 0) and c := (7 -// where /,7 € Fq such that /2 + 72 = -1. Then (a, c) is a representation of R in GL (2, q) (see [6, pp. 153-154]). Since R is irreducible on V, any R-orbit vR in V# contains a basis {v1, v2} of V, and vGo contains (v1)# U (v2)#. Clearly V# C (v1)# + (v2)#. Therefore V C vGo + vGo, and thus diam(r) = 2. This proves (4), and completes the proof of the proposition. □ 3.6 Class C7 In this case V = <8>f=1Uj with U = F^ for all i, m > 2, t > 2, and d = m4. Assume that B is a tensor product basis of V, with B := {<8>t=1Wij 11 < j < m . the C4 case, it is not d have As in the C4 case, it is not difficult to show that for any v = J21=1 (®j=1 vi,j) € V# we vT ^ (®j=iv,T,j), i=i where t acts on each Uj with respect to the basis {u^ | 1 < j < m}. 3.6.1 Case H = TL(n, q) By Theorem 2.4 G0 = (GL (m,q) fa Sym (t)) x (t>. (3.15) If t = 2 then we obtain the examples in Proposition 3.13 with k = m. We state this in the next corollary, which is analogous to [2, Corollary 3.7]. C. Amarra et al.: Affine primitive symmetric graphs of diameter two 163 Corollary 3.24. Let V = ®t=1Fm and let G0 be as in (3.15) with m > 2 and t = 2. Then the G0-orbits in V # are the sets YS for each s G {1,..., m}, where YS is as defined in (3.7). Moreover, for any G0 -orbit S Ç V #, the graph Cay(V, S ) has diameter 2 if and only if S = YS for some s > m/2. Proof. This follows immediately from Lemma 3.12 and Proposition 3.13. □ Using Lemma 2.1, we get the following bounds which significantly reduce the cases that remain to be considered. It turns out that these are exactly the same as those in [2, Proposition 3.8]; we prove them here for subgroups of rL(n, q). Proposition 3.25. Let r be a graph and let G < Aut (r), such that G satisfies Hypothesis 3.1 with G0 as in (3.15), m > 2 and t > 3. Then r is connected and G-symmetric if and only if r = Cay(V, vGo ) for some v G V#. Moreover, if diam(r) = 2 then either: 1. m = 2 and t G {3, 4, 5}; or 2. t = 3 and m G {3, 4, 5}. Proof. Recall that (agi ) g2 • • • gt = gi <8> • • • <8> (agi) • • • gt for all gi,..., gt G GL (m, q), so that |Go| < |GL (m,q) |t t! i(q - 1)-(t-1). Now |GL (m, q) | < qm(m—1)qm—1 (q - 1) = qm2-1(q - 1), s < qs-i for all s > 2 and q > 2, and i < = q for all i > 1 andp > 2, so that |Go|2 + 1 < (q(m2-1)t(q - 1)t)2 (q2t(t-1))2 q2(q - 1)-2(t-1) < qt2 + (2m2-3)t+4. It can be shown that t2 + (2m2 - 3)t + 4 < mt whenever t > 7 and m > 2, and whenever t g {3,4,5, 6} and m > m0 (t), where m0(t) is as given in Table 3.6.10. Hence |G0|2 + 1 < |V| for all such pairs (m,t). Of the remaining pairs we can eliminate (2, 6) and (6,3) by considering n(q, m, t) := (t!)2q2t(m -1)+4 - qmt; it can be shown that n(q, 2, 6) < 0 for all q > 2 and n(q, 6, 3) < 0 for all q > 7. For q G {2, 3,4,5} it can be checked that 36i2 |GL (6, q) |6(q - 1)-4 + 1 < q216. Therefore |Go|2 + 1 < |V| if (m, t) G {(2, 6), (6, 3)}, which completes the proof. □ Table 3.6.10: Values for m0(t) t 3 4 5 6" mo(t) 6 2 2 2 3.6.2 Case H = rSp(n, q) By Theorem 2.5, both q and t are odd and Go = (GSp (m, q) fa Sym (t)) x (t>. (3.16) Hence q, t > 3. 164 Ars Math. Contemp. 13 (2017) 107-123 Proposition 3.26. Let r be a graph and G < Aut (r) such that G satisfies Hypothesis 3.1 with G0 as in (3.16), m > 2 and t > 3. Then r is connected and G-symmetric if and only if r = Cay(V, vGo ) for some v G V#. Moreover, if diam(r) = 2 then either: 1. m = 2 and t G {3, 5}; or 2. t = 3, m = 4, and q = 9. Proof. In this case |Go| < |GSp (m, q) 1! £(q - 1)-(t-1), where |GSp (m, q) | = (q - 1)Sp (m, q) < (q - 1)q1 (m2+m). Also s < ks/2 for all k > 3 and s > 2, so that £ < q, t! < q4(t-1)(t+2), and |Go|2 + 1 < (q - 1)2iqi(m2+m)+1 (t-1)(t+2) + 1(q - 1)-2(t-1) < q2i2 + (m2+m+ 2)i+2. It can be shown that 112 + (m2 + m + 1) t + 2 < m4 whenever t > 6 and m > 2, t = 3 and m > 5, and t = 5 and m > 3. So |G012 + 1 < |V| for all such pairs (m, t). Let n(q, m, t) := (t!)2qt(m2+m)+3 - qm. If (m, t) = (4,3) then for all q > 37 we get |Go |2 + 1 -|V|