Vertex-imprimitive symmetric graphs with exactly one edge between any two distinct blocks
aa r X i v : . [ m a t h . G R ] J un Vertex-imprimitive symmetric graphs with exactly one edgebetween any two distinct blocks
Teng Fang , Xin Gui Fang , Binzhou Xia , Sanming Zhou Abstract
A graph Γ is called G -symmetric if it admits G as a group of automorphisms actingtransitively on the set of ordered pairs of adjacent vertices. We give a classification of G -symmetric graphs Γ with V (Γ) admitting a nontrivial G -invariant partition B such thatthere is exactly one edge of Γ between any two distinct blocks of B . This is achieved bygiving a classification of ( G, G -block-transitive designs D togetherwith G -orbits Ω on the flag set of D such that G σ,L is transitive on L \ { σ } and L ∩ N = { σ } for distinct ( σ, L ) , ( σ, N ) ∈ Ω, where G σ,L is the setwise stabilizer of L in the stabilizer G σ of σ in G . Along the way we determine all imprimitive blocks of G σ on V \ { σ } for every2-transitive group G on a set V , where σ ∈ V . Keywords : Symmetric graph; arc-transitive graph; flag graph; spread
Intuitively, a graph is symmetric if all its arcs have the same status in the graph, where an arc is an ordered pair of adjacent vertices. Since Tutte’s seminal work [28], symmetric graphs havelong been important objects of study in graph theory due to their intrinsic beauty and wideapplications (see [26] for an excellent overview of the area). In this paper we give a classificationof those symmetric graphs with an automorphism group acting transitively on the arc set andimprimitively on the vertex set such that there is exactly one edge between any two blocks ofthe underlying invariant partition.A finite graph Γ with vertex set V (Γ) is called G -symmetric (or G -arc-transitive ) if it admits G as a group of automorphisms (that is, G acts on V (Γ) and preserves the adjacency relationof Γ) such that G is transitive on V (Γ) and transitive on the set of arcs of Γ. (A graph is symmetric if it is Aut(Γ)-symmetric, where Aut(Γ) is the full automorphism group of Γ.) Thegroup G is said to be imprimitive on V (Γ) if V (Γ) admits a nontrivial G -invariant partition B = { B, C, . . . } , that is, 1 < | B | < | V (Γ) | and B g := { α g | α ∈ B } ∈ B for any g ∈ G and B ∈ B . In this case (Γ , G, B ) is said to be a symmetric triple . The quotient graph of Γ relative to B , denoted by Γ B , is defined to be the graph with vertex set B such that B, C ∈ B are adjacentif and only if there exists at least one edge of Γ with one end-vertex in B and the other in C .(As usual we assume that Γ B has at least one edge so that each block of B is an independent setof Γ.) For adjacent B, C ∈ B , define Γ[
B, C ] to be the bipartite subgraph of Γ with bipartition { Γ( C ) ∩ B, Γ( B ) ∩ C } , where Γ( B ) is the set of vertices of Γ with at least one neighbour in B .Since Γ B can be easily seen to be G -symmetric, this bipartite graph is independent of the choice Center for Applied Mathematics at Tianjin University, Tianjin 300072, P. R. China School of Mathematical Sciences, Peking University, Beijing 100871, P. R. China School of Mathematics and Statistics, University of Western Australia, Crawley 6009, WA, Australia School of Mathematics and Statistics, The University of Melbourne, Parkville, VIC 3010, Australia
E–mail addresses : [email protected] (T. Fang), [email protected] (X. G. Fang), [email protected] (B. Xia), [email protected] (S. Zhou).
1f adjacent
B, C up to isomorphism. Denote by Γ B ( B ) the neighbourhood of B in Γ B , and byΓ B ( α ) the set of blocks of B containing at least one neighbour of α ∈ V (Γ) in Γ. Denote v := | B | , r := | Γ B ( α ) | , b := | Γ B ( B ) | , k := | Γ( C ) ∩ B | . (1)Since Γ is G -symmetric and B is G -invariant, these parameters are independent of the choice of α ∈ V (Γ) and adjacent B, C ∈ B .Various possibilities for Γ[
B, C ] can happen. In the “densest” case where Γ[
B, C ] ∼ = K v,v is a complete bipartite graph, Γ is uniquely determined by Γ B , namely, Γ ∼ = Γ B [ K v ] is thelexicographic product of Γ B by the complete graph K v . The “sparsest” case where Γ[ B, C ] ∼ = K (that is, k = 1) can also happen; in this case Γ is called a spread of Γ B in [16], where it wasshown that spreads play a significant role in the study of edge-primitive graphs. Spreads havealso arisen from some other classes of symmetric graphs (see [24, 30, 31]), and a study of themwas undertaken in [32, Section 4]. Spreads of cycles and complete graphs with r = 1 were brieflydiscussed in [15, Section 4], where Gardiner and Praeger remarked that when k = 1 and Γ B isa complete graph “it is not at all clear what one can say about Γ in general”.In response to the remark above, in this paper we give a classification of all spreads ofcomplete graphs. Theorem 1.1.
All symmetric triples (Γ , G, B ) with G ≤ Aut(Γ) such that there is exactly oneedge of Γ between any two distinct blocks of B are classified in this paper and will be describedin Sections 3 and 4. Several interesting families of symmetric triples (Γ , G, B ) arise from this classification. Inparticular, we obtain four infinite families of connected symmetric spreads of complete graphs(see Lemmas 3.4, 3.7, 3.8 and 4.9). Such graphs are mutually non-isomorphic as they havedifferent orders or valencies.As shown in [15, Theorem 4.2], in the degenerate case where r = 1, we have Γ ∼ = (cid:0) v +12 (cid:1) · K (the graph of (cid:0) v +12 (cid:1) independent edges) and G can be any 2-transitive group of degree v + 1. Sowe will only consider the general case where r > G, G -block-transitive designs together with certain G -orbits on their flag sets.We give the latter classification in the following theorem, but postpone related definitions andresults on flag graphs to Section 2.2. (We use soc( G ) to denote the socle of a group G , that is,the product of its minimal normal subgroups. We use G to denote the stabilizer of the zerovector when G acts on a vector space V , and e = (1 , , . . . ,
0) the vector of V with 0 at everycoordinate except the first one.) Theorem 1.2.
Let D be a ( G, -point-transitive and G -block-transitive - ( | V | , r + 1 , λ ) ( r > design with point set V , where G ≤ Sym( V ) . Then there exists at most one 1-feasible G -orbit on the set of flags of D . Moreover, all possibilities for ( D , G ) such that such a 1-feasible G -orbit Ω exists, and the unique G -flag graph Γ( D , Ω , Ψ) associated with each ( D , G ) togetherwith its connectedness, are given in Tables 1-2, where Ψ is the set of ordered pairs of flags (( σ, L ) , ( τ, N )) ∈ Ω × Ω such that σ = τ and σ, τ ∈ L ∩ N . Furthermore, Ω is explicitly givenfor each ( D , G ) in all cases except (h)-(j) in Table 2. Theorem 1.1 follows from Theorem 1.2 and Corollary 2.2, but details of the correspondingflag graphs Γ( D , Ω , Ψ) will be given during the proof of Theorem 1.2. The number of pairs( D , G ) in each of (h)-(j) above will be computed by Magma but their structures will not begiven due to space limit. As a byproduct of the proof of Theorem 1.2, we give (or enumerate incases (h)-(j)) all imprimitive blocks of G σ on V \ { σ } for every 2-transitive group G on V .2 D Γ( D , Ω , Ψ) Details(a) A PG(3 ,
2) 35 · K § G ) = PSL( d, q ) PG( d − , q ) ( q d − q d − − q − q − · K q +1 § d ≥ G ≤ PΓU(3 , q ) 2-( q + 1 , q + 1 ,
1) ( q − q + q ) · K q +1 L3.2soc( G ) = PSU(3 , q ) q ≥ G ) = Sz( q ) 2-( q + 1 , q + 1 , q + 1) C, ord = q ( q + 1) L3.4 q = 2 e +1 > q (e) soc( G ) = R( q ) 2-( q + 1 , q + 1 ,
1) ( q − q + q ) · K q +1 § q = 3 e +1 ≥ G ) = R( q ) 2-( q + 1 , q + 1 , q + 1) C, ord = q ( q + 1) and L3.7 q = 3 e +1 ≥ q if q >
3; threecomponents if q = 3soc( G ) = R( q ) 2-( q + 1 , q + 1 , q + 1) C, ord = q ( q + 1) and L3.8 q = 3 e +1 ≥ q R(3) 2-(28 , ,
10) Three components L3.9Table 1. Theorem 1.2: almost simple case. Acronym: L = Lemma, T = Table, C = Connected,D = Disconnected, ord = Order, val = ValencyThe reader is referred to [24, 30, 31, 32] for several studies on vertex-imprimitive symmetricgraphs using the geometric approach developed in [15]. Together with [8, 17, 31], in [14] theauthors of the present paper completed the classification of symmetric triples (Γ , G, B ) such that k = v − ≥ B is a complete graph. The reader is referred to [11] and [4] for notation and terminology on permutation groups andblock designs, respectively. Unless stated otherwise, all designs are assumed to have no repeatedblocks, and each block of a design is identified with the set of points incident with it.Let G be a group acting on a set Ω. That is, for any α ∈ Ω and g ∈ G there correspondsa point in Ω denoted by α g , such that α G = α and ( α g ) h = α gh for any α ∈ Ω and g, h ∈ G ,where 1 G is the identity element of G . Let P i be a point or subset of Ω for i = 1 , , . . . , n . Define( P , P , . . . , P n ) g := ( P g , P g , . . . , P gn ) for g ∈ G , where P gi := { α g | α ∈ P i } if P i is a subset ofΩ. Denote P Gi := { P gi | g ∈ G } . In particular, α G is the G -orbit on Ω containing α . Define G P ,P ,...,P n := { g ∈ G | P gi = P i , i = 1 , . . . , n } ≤ G . In particular, if α is a point and P a subsetof Ω, then G α is the stabilizer of α in G , G P is the setwise stabilizer of P in G , and G α,P is thesetwise stabilizer of P in G α .Let G and H be groups acting on Ω and ∆, respectively. These two actions are said to be permutation isomorphic if there exist a bijection ρ : Ω → ∆ and an isomorphism η : G → H such that ρ ( α g ) = ( ρ ( α )) η ( g ) for α ∈ Ω and g ∈ G . If in addition G = H and η is the identityautomorphism of G , then the two actions are said to be permutation equivalent . It is immediatefrom the definition that if ϕ : G → Sym(Ω) and ψ : H → Sym(Ω) are monomorphisms, thenthe corresponding actions of G and H on Ω are permutation isomorphic if and only if ϕ ( G ) and ψ ( H ) are conjugate in Sym(Ω). Let Γ and Σ be G -symmetric graphs. If there exists a graphisomorphism ρ : V (Γ) → V (Σ) such that the actions of G on V (Γ) and V (Σ) are permutation3 D Γ( D , Ω , Ψ) Details(f) G ≤ AΓL(1 , q ) 2-( q, p t , D has a block L q ( q − p t ( p t − · K p t § q = p d such that L is a subfield of F q G ≤ AΓL(1 , q ) 2-( q, | L | , | L | ); D has a block C iff p ≡ − q = p d L with 0 , ∈ L and L \ { } d is odd, and P ≤ F × q the union of some cosets of with index 2; in this casea subgroup of F × q ord = 2 q and val = q − (g) G ≤ AΓL( n, q ) 2-( q n , | L | , λ ), λ = 1 or | L | ; D L4.10 G ☎ Sp( n, q ) D has a block L ⊆ h e i V = F nq , n ≥ { a ∈ F × q | a e ∈ L } a subgroup of F × q G ≤ AΓL( n, q ) As above D L4.11 G ☎ SL( n, q ) § V = F nq , n ≥ G ≤ AΓL(6 , q ) As above ( n = 6) D L4.13 G ☎ G ( q ) V = F q , q > G ≤ AGL(6 ,
3) 2-(3 , r + 1 , λ ), λ = 1 or r + 1; D T5 G ∼ = SL(2 , D has a block L such that V = F L \ { } is an orbit of some H ≤ G on V with G , x ≤ H for some x ∈ L \ { } (i) G ≤ AGL(2 , p ) 2-( p , r + 1 , λ ), λ = 1 or r + 1; D T8-14 G ☎ SL(2 ,
3) or D has a block L with the G ☎ SL(2 ,
5) same property as in (h) p = 5 , , , , , V = F p (j) G ≤ AGL(4 ,
3) 2-(3 , r + 1 , λ ), λ = 1 or r + 1; D T6-7 G ☎ SL(2 ,
5) or D has a block L with the G ☎ E , V = F same property as in (h)Table 2. Theorem 1.2: affine case. Acronym: L = Lemma, T = Table, C = Connected, D =Disconnected, ord = Order, val = Valency 4quivalent with respect to ρ , then Γ and Σ are said to be G -isomorphic with respect to the G -isomorphism ρ , and we denote this fact by Γ ∼ = G Σ. Let (Γ , G, B ) be a symmetric triple. As in [15], define D ( B ) := ( B, Γ B ( B )) to be the incidencestructure with “point set” B and “block set” Γ B ( B ) such that α ∈ B and C ∈ Γ B ( B ) areincident if and only if C ∈ Γ B ( α ). It can be verified [15] that D ( B ) is a 1-( v, k, r ) design with b blocks, and is independent of B up to isomorphism. Denote by D ∗ ( B ) := (Γ B ( B ) , B ) the dual1-design of D ( B ). We may identify the “blocks” α ∈ B of D ∗ ( B ) with the subsets Γ B ( α ) of the“point set” Γ B ( B ) of D ∗ ( B ), and we call two such “blocks” Γ B ( β ), Γ B ( γ ) repeated if β, γ ∈ B aredistinct but Γ B ( β ) = Γ B ( γ ). For α ∈ V (Γ), let B ( α ) denote the unique block of B containing α and set L ( α ) := { B ( α ) } ∪ Γ B ( α ). Let L be the set of all L ( α ), α ∈ V (Γ), with repeatedones identified. One can see that the action of G on B induces an action of G on L defined by L ( α ) g := L ( α g ), for α ∈ V (Γ) and g ∈ G . Define D (Γ , B ) := ( B , L )to be the incidence structure with incidence relation the set-theoretic inclusion. Then D (Γ , B )is a 1-design with block size r + 1 which admits G as a point- and block-transitive group ofautomorphisms ([32, Lemma 3.1]). In the case when D ∗ ( B ) has no repeated blocks (whichoccurs particularly when k = 1 by [32, Lemma 4.1(a)]), Γ can be reconstructed from D (Γ , B ) bythe following construction (see [31, 32] and [21, Construction 5.2]). Definition 2.1. ([32, Definitions 2.1 and 4.2]) Let D be a G -point-transitive and G -block-transitive 1-design with block size at least 2. Let σ be a point of D , Ω a G -orbit on the set offlags of D and Ω( σ ) the set of flags of Ω with point entry σ . The set Ω is said to be feasible if(a) | Ω( σ ) | ≥
2; and(b) for some (and hence all) flag ( σ, L ) ∈ Ω, G σ,L is transitive on L \ { σ } .If Ω also satisfies(c) L ∩ N = { σ } , for distinct ( σ, L ) , ( σ, N ) ∈ Ω( σ ),then Ω is said to be .Given a feasible G -orbit Ω on the set of flags of D , denoteΞ( D , Ω) := { (( σ, L ) , ( τ, N )) ∈ Ω × Ω | σ = τ and σ, τ ∈ L ∩ N } . (2)If Ψ is a self-paired G -orbit (that is, (( σ, L ) , ( τ, N )) ∈ Ψ implies (( τ, N ), ( σ, L )) ∈ Ψ) on Ξ( D , Ω),then the G -flag graph of D with respect to (Ω , Ψ), denoted by Γ( D , Ω , Ψ), is defined to be thegraph with vertex set Ω and arc set Ψ.The following was proved in [32, Theorem 1.1]: If D ∗ ( B ) contains no repeated blocks, thenΓ is G -isomorphic to a G -flag graph of D (Γ , B ) with respect to some (Ω , Ψ). Conversely, any G -flag graph Γ( D , Ω , Ψ) is a G -symmetric graph admitting B (Ω) := { Ω( σ ) | σ is a point of D} as a G -invariant partition such that the corresponding D ∗ (Ω( σ )) contains no repeated blocks.The case where k = 1 occurs if and only if Ω is 1-feasible, and in this case G is faithful on V (Γ) if and only if it is faithful on the point set of D (see [32, Theorem 4.3] and the remarkbelow it). In the case where k = 1 and Γ B is a complete graph, we have Γ B ∼ = K vr +1 as Γ B has5alency vr . Since Γ B is G -symmetric, this occurs precisely when G is 2-transitive on B . Hencein this case D (Γ , B ) is a ( G, G -block-transitive 2-( vr + 1 , r + 1 , λ ) designfor some integer λ ≥
1. Conversely, if D is a ( G, G -block-transitive 2-( vr + 1 , r + 1 , λ ) design, then for any G -flag graph Γ = Γ( D , Ω , Ψ) of D , we have Γ B (Ω) ∼ = K vr +1 .Thus [32, Theorem 4.3] has the following consequence. Corollary 2.2. ([32, Corollary 4.4]) Let v ≥ and r ≥ be integers, and let G be a group.Then the following statements are equivalent: (a) Γ is a G -symmetric graph of valency r admitting a nontrivial G -invariant partition B ofblock size v such that Γ[ B, C ] ∼ = K for any two distinct blocks B, C of B ( so Γ B ∼ = K vr +1 ) ; (b) Γ ∼ = G Γ( D , Ω , Ψ) , for a ( G, -point-transitive and G -block-transitive - ( vr + 1 , r + 1 , λ ) design D , a 1-feasible G -orbit Ω on the set of flags of D , and a self-paired G -orbit Ψ on Ξ( D , Ω) .In addition, G is faithful on V (Γ) if and only if it is faithful on the point set of D .Moreover, for any point σ of D , the set of points of D other than σ admits a G σ -invariantpartition of block size r , namely, { L \ { σ } | ( σ, L ) ∈ Ω } . Hence D is not ( G, -point-transitivewhen r ≥ . Furthermore, either λ = r + 1 , or λ = 1 and Γ ∼ = ( v ( vr + 1) / ( r + 1)) · K r +1 . In the case where r = 1, by the G -flag graph construction, V (Γ) can be identified with V (2) := { ( i, j ) | ≤ i, j ≤ v, i = j } . Thus B = { B , B , B , . . . , B v } where B i = { ( i, j ) | ≤ j ≤ v, j = i } , Γ = (cid:0) v +12 (cid:1) · K with edge set {{ ( i, j ) , ( j, i ) } | ≤ i, j ≤ v, i = j } , and G is any2-transitive group on { , , , . . . , v } acting on V (2) coordinate-wise (see also [15, Theorem 4.2]).In what follows we assume r >
1. Since the graphs in Theorem 1.1 are precisely those inCorollary 2.2 (a), to prove Theorem 1.1 it suffices to classify ( D , Ω , Ψ) in part (b) of this corollarywith G faithful on the point set of D . In the remainder of the paper we give this classificationand thus prove Theorem 1.2. Lemma 2.3.
Let D be a ( G, -point-transitive and G -block-transitive - ( | V | , r + 1 , λ ) designwith point set V . Then there is at most one 1-feasible G -orbit on the flag set of D . Moreover,if such an orbit exists, say Ω = ( δ, L ) G , then either (a) G L is transitive on L ( or equivalently G L G δ ) , λ = 1 , and Ω is the set of all flags of D ; or (b) G L is not transitive on L ( orequivalently G L ≤ G δ ) and λ = r + 1 . Proof.
Let σ, τ ∈ V be distinct points. Denote by L , . . . , L λ the λ blocks of D containing both σ and τ . Denote Ω i := ( σ, L i ) G , i = 1 , . . . , λ. Since G is 2-transitive on V , the sets Ω , . . . , Ω λ are all possible G -orbits on the flag set of D ,possibly with Ω i = Ω j for distinct i and j . Suppose that Ω = ( δ, L ) G is 1-feasible. Then by (c)in Definition 2.1 we have | V | = vr + 1 for some integer v .First assume that G L is transitive on L . Then ( η, L ) ∈ Ω for every η ∈ L . Hence, by thetransitivity of G on the block set of D , Ω = Ω = · · · = Ω λ is the flag set of D and thus λ = 1by (c) in Definition 2.1.Next assume that G L is not transitive on L . Then G L ≤ G δ by (b) in Definition 2.1.Let η be a fixed point of V . For each π ∈ V \ { η } , by (c) in Definition 2.1 there is onlyone flag in Ω( π ) whose block entry contains η . On the other hand, if ( τ , M ) and ( τ , M )are flags in Ω for distinct τ , τ ∈ V and some M ∈ L G , then τ , τ ∈ M and there exists g ∈ G such that ( τ , M ) = ( τ , M ) g . Thus g ∈ G M and τ = τ g . Since Ω satisfies (b) in6efinition 2.1, G M is transitive on M , but this contradicts our assumption. Hence there are | Ω( η ) | + ( | V | −
1) = v + rv = ( r + 1) v blocks of D containing η . By the relations betweenparameters of the 2-design D , we get λ = r + 1.Suppose that Ω i = Ω j and both of them are 1-feasible. Since D is G -block-transitive, thereexists a point ξ of D such that ( ξ, L j ) ∈ Ω i . The assumption Ω i = Ω j implies σ = ξ and G L j = G ξ,L j ≤ G ξ (for otherwise G L j is transitive on L j and Ω i = Ω j is the flag set of D bythe proof above). Since ξ ∈ L j \ { σ } , G σ,L j ≤ G L j ≤ G ξ and | L j | = r + 1 ≥ G σ,L j cannot betransitive on L j \ { σ } , which contradicts the assumption that Ω j is 1-feasible. Hence there is atmost one 1-feasible G -orbit on the flag set of D . Lemma 2.4.
Let D be a ( G, -point-transitive and G -block-transitive - ( | V | , r + 1 , λ ) designwith point set V . Suppose that there is a 1-feasible G -orbit Ω = ( σ, L ) G on the flag set of D . Let P := L \ { σ } and let H be a transitive subgroup of G σ on V \ { σ } . Then Ξ( D , Ω) (see (2) ) is aself-paired G -orbit on the set of ordered pairs of distinct flags in Ω , P is an imprimitive block of H on V \ { σ } , and P is the union of some H τ -orbits (including the H τ -orbit { τ } of length 1),where τ ∈ P is a fixed point. Proof.
By (c) in Definition 2.1 we know that P = { M \ { δ } | ( δ, M ) ∈ Ω } is a G σ -invariantpartition of V \ { σ } . Since H ≤ G σ is transitive on V \ { σ } , it follows that P is an imprimitiveblock of H on V \ { σ } . Since τ ∈ P , P is H τ -invariant and so is the union of some H τ -orbits.Let (( δ, M ) , ( π, N )), (( ξ, M ) , ( η, N )) ∈ Ξ( D , Ω). Then there exists g ∈ G such that δ g = ξ and π g = η . Hence ξ, η ∈ M g ∩ N g . By (c) in Definition 2.1, we have M = M g and N = N g , andthus Ξ( D , Ω) is a self-paired G -orbit on the set of ordered pairs of distinct flags in Ω.From now on we use the following abbreviations when D and Ω are clear from the context:Ξ := Ξ( D , Ω) , Γ( D , Ω , Ξ) := Γ( D , Ω , Ξ( D , Ω)) . Given a group G , a subgroup T of G , and an element g ∈ G with g N G ( T ) and g ∈ T ∩ T g ,define the coset graph Cos( G, T, T gT ) to be the graph with vertex set [ G : T ] := { T x | x ∈ G } and edge set {{ T x, T y } | xy − ∈ T gT } . It is well known (see e.g. [25]) that Cos( G, T, T gT ) isa G -symmetric graph with G acting on [ G : T ] by right multiplication, and Cos( G, T, T gT ) isconnected if and only if h T, g i = G . Conversely, any G -symmetric graph Γ is G -isomorphic toCos( G, T, T gT ) (see e.g. [25]), where g is an element of G interchanging two adjacent vertices α and β of Γ and T := G α , and the required G -isomorphism is given by V (Γ) → [ G : T ] , γ T x ,with x ∈ G satisfying α x = γ . Based on this one can prove the following result. Lemma 2.5.
Let (( σ, L ) , ( τ, N )) ∈ Ξ and T := G σ,L . Let g ∈ G satisfy ( σ, τ ) g = ( τ, σ ) and set H := h T, g i . Then ρ : Ω → [ G : T ] , γ T x , with x ∈ G satisfying ( σ, L ) x = γ ,defines a G -isomorphism from Γ( D , Ω , Ξ) to Cos(
G, T, T gT ) , under which the preimage of thesubgraph Cos(
H, T, T gT ) of Cos(
G, T, T gT ) is the connected component of Γ( D , Ω , Ξ) containingthe vertex ( σ, L ) . In the proof of Theorem 1.2, we will use the following results whose proofs are straightforwardand hence omitted.
Lemma 2.6.
Let ℓ be a nonnegative integer and q > a prime power such that | ( q + 1) . (a) If ( ℓ ( q − / q ) | q , then ℓ = 0 or q ; (b) if ( ℓ ( q − / | q , then ℓ = 0 or . Lemma 2.7.
Let ℓ and n be positive integers and q > a prime power. If ( ℓ ( q −
1) + 1) | q n ,then ℓ = ( q i − / ( q − for some i = 1 , , . . . , n . Lemma 2.8.
Let ℓ ≥ be an integer, n a positive integer, and q an odd power of . Then ( ℓ ( q −
1) + ( q − / ∤ q n . .4 Methodology Let D be a ( G, G -block-transitive 2-( | V | , r + 1 , λ ) ( r >
1) design withpoint set V , where G ≤ Sym( V ). By Lemma 2.4, for a 1-feasible G -orbit Ω = ( σ, L ) G on the setof flags of D , L \ { σ } must be an imprimitive block of G σ on V \ { σ } . The major task in theproof of Theorem 1.2 is to determine such blocks by examining subgroups H of G σ , with thehelp of the following methods.(i) Suppose that H ≤ G σ is transitive on V \ { σ } . For each nontrivial imprimitive block P of H on V \ { σ } , we will check whether P is also an imprimitive block of G σ on V \ { σ } . ByLemma 2.4, P is the union of some H τ -orbits on V \ { σ } , where τ ∈ P .(ii) If there is a point τ ∈ V \ { σ } such that H τ = G σ,τ , then by [11, Theorem 1.5A], P := τ H is an imprimitive block of G σ on V \ { σ } .If P is an imprimitive block of G σ on V \ { σ } obtained from (i) or (ii), then define D := ( V, L G ) , where L := P ∪ { σ } , to be the incidence structure with point set V and block set L G , with incidence relation theset-theoretic inclusion. By [4, Proposition 4.6], D is a 2-( | V | , | P | + 1 , λ ) design admitting G asa 2-point-transitive and block-transitive group of automorphisms. Moreover, Ω := ( σ, L ) G isthe unique 1-feasible G -orbit on the flag set of D , and ( σ, L ) is adjacent to r := | P | vertices( η , M ), . . . , ( η r , M r ) in the G -flag graph Γ( D , Ω , Ξ), where { η , . . . , η r } = P and σ ∈ M i \ { η i } , i = 1 , , . . . , r . The value of λ and the connectedness of Γ( D , Ω , Ξ) will be determined for each P . Since G is 2-transitive on V , it is either almost simple (with socle a nonabelian simple group)or affine (with socle an abelian group). We will deal with these two cases in the next two sections,with results summarized in Tables 1 and 2, respectively. In this section we assume that G ≤ Sym( V ) is 2-transitive on V of degree u := | V | with soc( G ) anonabelian simple group. It is well known ([22], [7, p.196], [6]) that soc( G ) and u are as follows:(i) soc( G ) = A u , u ≥ G ) = PSL( d, q ), d ≥ q is a prime power and u = ( q d − / ( q − d, q ) =(2 , , (2 , G ) = PSU(3 , q ), q ≥ u = q + 1;(iv) soc( G ) = Sz( q ), q = 2 e +1 > u = q + 1;(v) soc( G ) = R( q ) ′ , q = 3 e +1 and u = q + 1;(vi) G = Sp d (2), d ≥ u = 2 d − ± d − ;(vii) G = PSL(2 , u = 11;(viii) soc( G ) = M u , u = 11 , , , , G = M , u = 12;(x) G = A , u = 15;(xi) G = HS, u = 176; 8xii) G = Co , u = 276.Let σ, τ be distinct points in V . In cases (i), (viii) and (ix), since soc( G ) is 3-transitive, a2-design as in Lemma 2.4 admitting G as a group of automorphisms does not exist. In case (vii),since G σ,τ has orbit-lengths 3 and 6 on V \ { σ, τ } (see [22]), again a 2-design as in Lemma 2.4does not exist. In case (xi), since G σ,τ has orbit-lengths 12, 72 and 90 on V \ { σ, τ } (see [22]),there is no 2-design as in Lemma 2.4. Similarly, in case (xii) such a 2-design does not exist as G σ,τ has orbit-lengths 112 and 162 on V \ { σ, τ } (see [22]).In case (ii) with d = 2, let σ = h e i and τ = h e i , where e = (1 ,
0) and e = (0 , ϕ ∈ PSL(2 , q ) σ,τ , then e ϕ = ( µ,
0) and e ϕ = (0 , /µ ) for some µ ∈ F × q . Let η = h ( x, i ∈ V \ { σ, τ } ,where x ∈ F × q . Then η ϕ = h ( xµ , i and the PSL(2 , q ) σ,τ -orbit on V containing η has length q − q is even and ( q − / q is odd. Thus, if P is a nontrivial imprimitive blockof PSL(2 , q ) σ on V \ { σ } containing τ , then q is odd and | P | = 1 + ( q − / q + 1) /
2. Since | P | divides u − q , we must have q = 1, a contradiction. Hence this case does not produceany G -flag graph.In case (ii) with d ≥ G σ,τ has orbit-lengths q − u − ( q + 1) on V \ { σ, τ } . Thus D = PG( d − , q ) and λ = 1 by [22]. The imprimitive block of G σ on V \ { σ } containing τ hassize q and so Γ( D , Ω , Ξ) ∼ = (( q d − q d − − / ( q − q − · K q +1 , contributing to (b) in Table1. In case (vi), G σ acts on V \ { σ } as O ± (2 d,
2) does on its singular vectors (see [22]), and G σ,τ has orbit-lengths 2(2 d − ∓ d − ±
1) and 2 d − on V \ { σ, τ } . (We may assume d ≥ d < G σ,τ -orbit on V \ { σ, τ } plus 1 cannot divide u −
1, there is no 2-design as in Lemma2.4 admitting G as a group of automorphisms.In case (x), since G σ,τ has orbit-lengths 1 and 12 on V \ { σ, τ } , we have D = PG(3 ,
2) and λ = 1 (see [22]). Since the imprimitive block of G σ on V \ { σ } containing τ has size 2, we haveΓ( D , Ω , Ξ) ∼ = 35 · K , contributing to (a) in Table 1. soc( G ) = PSU(3 , q ) , q ≥ a prime power and u = q + 1 This case yields (c) in Table 1. We use the following permutation representation of PSU(3 , q )(see [11, pp.248–249]). Let W be a 3-dimensional vector space over F q . Using ξ ξ = ξ q todenote the automorphism of F q of order 2, we define a hermitian form ϕ : W × W → F q by ϕ ( w, z ) = ξ η + ξ η + ξ η , where w = ( ξ , ξ , ξ ) and z = ( η , η , η ) are vectors in W . Onecan see that for this form the set of 1-dimensional isotropic subspaces is given by V = {h (1 , , i} ∪ {h ( α, β, i | α + α + ββ = 0 , α, β ∈ F q } . (A vector w ∈ W is isotropic if ϕ ( w, w ) = 0.) We have | V | = q + 1 and PSU(3 , q ) is 2-transitiveon V . Let π : GU(3 , q ) → PGU(3 , q ) = GU(3 , q ) /Z (GU(3 , q )) be the natural homomorphism,where Z (GU(3 , q )) is the center of GU(3 , q ). Denote t α,β := − β α β , h γ,δ := γ δ
00 0 γ − , where α, β, γ, δ ∈ F q . If δδ = 1, γ = 0 and α + α + ββ = 0, then t α,β and h γ,δ define elements of GU(3 , q ). There are q matrices of type t α,β and ( q − q + 1) of type h γ,δ . Let e = (1 , ,
0) and e = (0 , , , q ) h e i = { π ( h γ,δ ) π ( t α,β ) | α, β, γ, δ ∈ F q , δδ = 1 , γ = 0 , α + α + ββ = 0 } andGU(3 , q ) h e i , h e i = { h γ,δ | γ, δ ∈ F q , δδ = 1 , γ = 0 } . Obviously t α,β ∈ SU(3 , q ), and h γ,δ ∈ SU(3 , q ) if and only if δ = γ q − . Moreover, h γ,δ ∈ SU(3 , q ) is a scalar matrix if and only if γ q − = 1.In the rest of this section we denote PSU(3 , q ) by J .9 emma 3.1. ([14, Lemma 3.2]) Let h ( η , η , i ∈ V \ {h e i , h e i} . Denote by Q the J h e i , h e i -orbit containing h ( η , η , i . If η = 0 , then | Q | = q − ; if η = 0 , then | Q | = | J h e i , h e i | = ( q − , if ∤ ( q + 1) , ( q − / , if | ( q + 1) . Suppose that P is a nontrivial imprimitive block of J h e i on V \ {h e i} containing h e i . Then P \ {h e i} is the union of some J h e i , h e i -orbtis on V \ {h e i , h e i} . By Lemmas 2.6 and 3.1, wehave | P | = q or q . Lemma 3.2.
Let P be a nontrivial imprimitive block of J h e i on V \ {h e i} containing h e i .Then P = {h ( α, , i | α + α = 0 } . Moreover, let D := ( V, L J ) and Ω := ( h e i , L ) J , where L := P ∪ {h e i} , and let G ≤ PΓU(3 , q ) with soc( G ) = J . Then D is a - ( q + 1 , q + 1 , designadmitting G as a group of automorphisms, Ω is a 1-feasible G -orbit on the flag set of D , and Γ( D , Ω , Ξ) ∼ = ( q − q + q ) · K q +1 . Proof.
We first prove that | P | 6 = q . Suppose otherwise. Denote the q solutions in F q of theequation x + x = 0 by ε = 0, ε , . . . , ε q − . Then h ( ε , , i , . . . , h ( ε q − , , i is a J h e i , h e i -orbiton V \{h e i , h e i} . By Lemma 2.6, h ( ε i , , i is not contained in P for i >
0. Now Σ := { P g | g ∈ J h e i } is a system of blocks of J h e i on V \{h e i} with | Σ | = q , and T := h π ( t α,β ) | α + α + ββ = 0 i is transitive on Σ. Actually, T is a normal subgroup of J h e i acting regularly on V \ {h e i} (see[11, p.249]). Thus, the stabilizer of P in T has order q , that is, | T P | = q . For π ( t α ,β ), π ( t α ,β ) ∈ T P , we have h (0 , , i π ( t α ,β ) π ( t − α ,β ) = h ( α , β, i π ( t − α − ββ, − β ) = h ( α − α , , i ∈ P .Since h ( ε i , , i is not contained in P for i >
0, we have α = α and π ( t α ,β ) = π ( t α ,β ).Therefore, { β | h ( α, β, i ∈ P } = { β | π ( t α,β ) ∈ T P } = F q . (3)For any h ( η , η , i , h ( ξ , ξ , i ∈ P , η , ξ = 0, since P is an imprimitive block of J h e i on V \ {h e i} , both π ( t η ,η ) and π ( t ξ ,ξ ) fix P setwise. Thus h (0 , , i π ( t η ,η ) π ( t ξ ,ξ ) = h ( η , η , i π ( t ξ ,ξ ) = h ( η + ξ − ξ η , η + ξ , i ∈ P, h (0 , , i π ( t ξ ,ξ ) π ( t η ,η ) = h ( ξ , ξ , i π ( t η ,η ) = h ( ξ + η − η ξ , ξ + η , i ∈ P. Hence η + ξ − ξ η = ξ + η − η ξ by (3), that is, ( ξ /η ) q − = 1, which implies ( ξ /η ) ∈ Fix f ( F q ), where f denotes the automorphism of F q of order 2. Choosing η = 1, we have ξ ∈ Fix f ( F q ) and so F q ⊆ Fix f ( F q ), a contradiction. Thus | P | 6 = q .So we have | P | = q . By Lemma 3.1, P = {h ( α, , i | α + α = 0 } . Since G ≤ PΓU(3 , q ), P is also an imprimitive block of G h e i on V \ {h e i} , and moreover Ω = ( h e i , L ) G and L G = L J .Hence D is a 2-( q + 1 , q + 1 , λ ) design admitting G as a group of automorphisms with λ = 1(see [11, p.249] and [22]), Ω is a 1-feasible G -orbit on the flag set of D , and Γ( D , Ω , Ξ) ∼ =( q − q + q ) · K q +1 by Corollary 2.2. soc( G ) = Sz( q ) , q = 2 e +1 > and u = q + 1 This case yields (d) in Table 1. We use the following permutation representation of Sz( q ) (see[11, p.250]). The mapping σ : ξ ξ e +1 is an automorphism of F q and σ is the Frobeniusautomorphism ξ ξ . Define V := { ( η , η , η ) ∈ F q | η = η η + η σ +21 + η σ } ∪ {∞} . (4)Then | V | = q + 1. For α, β, κ ∈ F q with κ = 0, define the following permutations of V fixing ∞ : t α,β : ( η , η , η ) ( η + α, η + β + α σ η , µ ) ,n κ : ( η , η , η ) ( κη , κ σ +1 η , κ σ +2 η ) , µ = η + αβ + α σ +2 + β σ + αη + α σ +1 η + βη . Define the involution w fixing V by w : ( η , η , η ) ↔ (cid:18) η η , η η , η (cid:19) for η = 0 , ∞ ↔ (0 , ,
0) =: . Then Sz( q ) is generated by w and all t α,β and n κ . We have Sz( q ) ∞ = h t α,β , n κ | α, β, κ ∈ F q , κ = 0 i and Sz( q ) ∞ , = h n κ | κ ∈ F q , κ = 0 i , the latter being a cyclic group. Lemma 3.3.
Every
Sz( q ) ∞ , -orbit on V \ {∞ , } has length q − . Proof.
Since 2 e +1 + 1 and 2 e +1 − F × q is a cyclic group of order 2 e +1 − F × q → F × q , z z σ +1 is a group automorphism. Thus, if η = 0 or η = 0, then( aη , a σ +1 η , a σ +2 η ) = ( bη , b σ +1 η , b σ +2 η ) if and only if a = b , and so the result follows. Lemma 3.4.
Let G be any subgroup of Sym( V ) containing Sz( q ) as a normal subgroup. Supposethat P is a nontrivial imprimitive block of G ∞ on V \ {∞} containing . Then the followinghold: (a) P = { (0 , η, η σ ) ∈ V | η ∈ F q } and Sz( q ) ∞ ,P = h t ,ξ , n κ | κ ∈ F × q , ξ ∈ F q i ([14, Lemma 3.6]); (b) setting L := P ∪ {∞} , D := ( V, L
Sz( q ) ) = ( V, L G ) is a - ( q + 1 , q + 1 , q + 1) designadmitting G as a -point-transitive and block-transitive group of automorphisms, and Ω :=( ∞ , L ) Sz( q ) = ( ∞ , L ) G is a 1-feasible G -orbit on the flag set of D ; moreover, the G -flaggraph of D with respect to (Ω , Ξ) is the same as the Sz( q ) -flag graph Γ( D , Ω , Ξ) ; (c) Γ( D , Ω , Ξ) is connected with order | Ω | = q ( q + 1) and valency q . Proof.
We prove (b) and (c) only, as (a) was proved in [14, Lemma 3.6]. Since Sz( q ) is 2-transitive on V , D is a 2-( q + 1 , q + 1 , λ ) design admitting Sz( q ) as a 2-point-transitive andblock-transitive group of automorphisms. Since w does not stabilize L , λ = 1 and thus λ = q + 1by Lemma 2.3.Since Sz( q ) is a normal subgroup of G and Sz( q ) has index 2 e + 1 in its normalizer Q inSym( V ) ([7, p.197, Table 7.4]), Q/ Sz( q ) is a cyclic group of order 2 e + 1 and G = h Sz( q ) , ζ i ,where ζ is an automorphism of F q inducing a permutation of V that fixes ∞ and acts on theelements of V \ {∞} componentwise. Hence P = { (0 , η, η σ ) ∈ V | η ∈ F q } is a nontrivialimprimitive block of G ∞ on V \ {∞} . Moreover, ( ∞ , L ) G = ( ∞ , L ) Sz( q ) and L G = L Sz( q ) . Hence D admits G as an automorphism group and Ω is a 1-feasible G -orbit on the flag set of D .Denote H := h t ,ξ , w | ξ ∈ F q i . For any ( η , η , η ) ∈ V \ {∞ , } , if η = 0 then t ,η =( η , η , η ), and if η = 0 then t ,θ wt ,η = ( η , η , η ), where θ/θ σ = η . Hence H is transitiveon V , and thus ( q + 1) q divides | H | . So | H | does not divide q ( q − q − q + √ q + 1)or 4( q − √ q + 1). Thus, by [27, p.137, Theorem 9], | H | = ( s + 1) s ( s − s m = q forsome positive integer m . It follows that m = 1, | H | = ( q + 1) q ( q − q ) = H .Therefore, Sz( q ) = h Sz( q ) ∞ ,P , w i and so Γ( D , Ω , Ξ) is connected by Lemma 2.5. soc( G ) = R( q ) , q = 3 e +1 > , u = q + 1 ; or G = R(3) , R(3) ′ ∼ = PSL(2 , , u = 28 This case yields (e) in Table 1. We use the following permutation representation of R( q ) (see[11, p.251]). The mapping σ : ξ ξ e +1 is an automorphism of F q and σ is the Frobeniusautomorphism ξ ξ . The set V of points on which R( q ) acts consists of ∞ and the set of6-tuples ( η , η , η , λ , λ , λ ) with η , η , η ∈ F q and λ = η η − η η + η σ − η σ +31 ,λ = η σ η σ − η σ + η η + η η − η σ +31 ,λ = η η σ − η σ +11 η σ + η σ +31 η + η η − η σ +12 − η + η σ +41 . (5)11hus | V | = q + 1. For α, β, γ, κ ∈ F q with κ = 0, define the following permutations of V fixing ∞ : t α,β,γ : ( η , η , η , λ , λ , λ ) ( η + α, η + β + α σ η , η + γ − αη + βη − α σ +1 η , µ , µ , µ ) ,n κ : ( η , η , η , λ , λ , λ ) ( κη , κ σ +1 η , κ σ +2 η , κ σ +3 λ , κ σ +3 λ , κ σ +4 λ ) , where µ , µ , µ can be calculated from the formulas in (5). Define the involution w fixing V by w : ( η , η , η , λ , λ , λ ) ↔ (cid:18) λ λ , λ λ , η λ , η λ , η λ , λ (cid:19) for λ = 0, ∞ ↔ (0 , , , , ,
0) =: . (We use the corrected definition [12] of the action of w on V [11, p.251].) The Ree group R( q )is the group generated by w and all t α,β,γ and n κ . We have R( q ) ∞ = h t α,β,γ , n κ | α, β, γ, κ ∈ F q , κ = 0 i and R( q ) ∞ , is the cyclic group h n κ | κ ∈ F q , κ = 0 i . Since the first three coordinatesin each element of V determine the other three, we present an element of V by ( η , η , η , . . . ).The following lemma was used in [14], and its proof is given below for later reference. Lemma 3.5. ([14, Lemma 3.10]) Let ( η , η , η , . . . ) ∈ V \ {∞ , } . Then | ( η , η , η , . . . ) R( q ) ∞ , | = ( q − , if η = 0 or η = 0 , ( q − / , if η = η = 0 . Proof.
Since id : F × q → F × q , ξ ξ and ϕ : F × q → F × q , ξ ξ σ +2 are both group automorphisms,if η = 0 or η = 0, then ( aη , a σ +1 η , a σ +2 η , . . . ) = ( bη , b σ +1 η , b σ +2 η , . . . ) if and only if a = b .Let δ be a generator of the cyclic group F × q . Since δ σ +1 = δ e +1 +1 and gcd(3 e +1 +1 , q −
1) = 2,we have | δ σ +1 | = ( q − /
2, and thus L := (0 , , , . . . ) R( q ) ∞ , and L := (0 , δ, , . . . ) R( q ) ∞ , (6)are R( q ) ∞ , -orbits on V \ {∞ , } of length ( q − / q ) ∞ , has two orbits of length ( q − / q ( q + 1) orbits of length q − V \ {∞ , } .In the remainder of this section, let G ≤ Sym( V ) contain R( q ) as a normal subgroup and P be a nontrivial imprimitive block of G ∞ on V \ {∞} containing . Since R( q ) has index 2 e + 1in its normalizer Q in Sym( V ) ([7, p.197, Table 7.4]), Q/ R( q ) is a cyclic group of order 2 e + 1and G = h R( q ) , ζ i , (7)where ζ is an automorphism of F q inducing a permutation of V that fixes ∞ and acts on theelements of V \ {∞} componentwise. We have G ∞ = h R( q ) ∞ , ζ i , G ∞ , = h R( q ) ∞ , , ζ i . (8)By Lemma 2.4, P \ { } is the union of some R( q ) ∞ , -orbits on V \ {∞ , } . By Lemmas 2.7and 2.8, we have | P | = q or | P | = q . If | P | = q , then by Lemma 2.8, either L ∪ L ⊆ P or( L ∪ L ) ∩ P = ∅ , where L and L are as defined in (6). Case (i): | P | = q , P \ { } = L ∪ L Now P = { (0 , η, , . . . ) | η ∈ F q } . Since G ∞ = h t α,β,γ , n κ , ζ | α, β, γ ∈ F q , κ ∈ F × q i by (8) and(0 , η, , . . . ) t α,β,γ = ( α, η + β, γ − αη, . . . ), P is an imprimitive block of G ∞ on V \ {∞} . Denote12 := P ∪ {∞} , and define D := ( V, L G ) = ( V, L R( q ) ) and Ω := ( ∞ , L ) G = ( ∞ , L ) R( q ) . Then D is a 2-( q + 1 , q + 1 , λ ) design admitting G as a group of automorphisms, and Ω is a 1-feasible G -orbit on the flag set of D . It can be verified that w stabilizes L and thus λ = 1. It followsthat Γ( D , Ω , Ξ) ∼ = ( q − q + q ) · K q +1 , yielding the first possibility in (e) in Table 1. Case (ii): | P | = q , P \ { } is an R( q ) ∞ , -orbit on V \ {∞ , } of length q − P is of the form ( κη , κ σ +1 η , κ σ +2 η , . . . ) for some κ ∈ F q , where( η , η , η , . . . ) is a fixed point in P \ { } . Suppose that P ∩ P t α,β,γ = ∅ , that is, for some κ , κ ∈ F q , ( κ η , κ σ +11 η , κ σ +21 η , . . . ) t α,β,γ = ( κ η , κ σ +10 η , κ σ +20 η , . . . ) . (9)Then κ η + α = κ η , κ σ +11 η + β + α σ κ η = κ σ +10 η and κ σ +21 η + γ − ακ σ +11 η + βκ η − α σ +1 κ η = κ σ +20 η , or equivalently, α = ( κ − κ ) η ,β = ( κ σ +10 − κ σ +11 ) η − ( κ σ − κ σ ) κ η σ +11 ,γ = ( κ σ +20 − κ σ +21 ) η + ( κ κ σ +11 − κ κ σ +10 ) η η + ( κ σ − κ σ ) κ κ η σ +21 . (10)Hence, if α, β, γ are given by (10) in terms of η , η , η , then (9) holds. Since P is an imprimitiveblock of R( q ) ∞ on V \ {∞} , we need to verify that P t α,β,γ = P , that is, for every ℓ ∈ F q , thefollowing equation system has a solution x ∈ F q : ( x − ℓ ) η = α, ( x σ +1 − ℓ σ +1 ) η − ( x σ − ℓ σ ) lη σ +11 = β, ( x σ +2 − ℓ σ +2 ) η + ( xℓ σ +1 − ℓx σ +1 ) η η + ( x σ − ℓ σ ) xℓη σ +21 = γ. (11) Lemma 3.6. If (11) has a solution for every ℓ ∈ F q , then η = η = 0 . Proof. If P t η,θ,ξ ∩ P = ∅ for any t η,θ,ξ = id, then different t η,θ,ξ must map P to different elementsin P R( q ) ∞ , and thus q = |h t η,θ,ξ | η, θ, ξ ∈ F q i| ≤ | P R( q ) ∞ | = q , a contradiction. Hence wecan assume that at most two of α, β and γ are 0 in (9). We claim that η = 0. Supposeotherwise. Then x = α/η + ℓ , α = 0 (for otherwise x = ℓ and β = γ = 0 by (11)), and thesecond equation of (11) becomes αη η ℓ σ + (cid:16) α σ η η σ − α σ η (cid:17) ℓ + α σ +1 η η σ +11 − β = 0, which holds forevery ℓ ∈ F q . On the other hand, the polynomial of ℓ on the left-hand side should have at most3 e +1 roots if it is nonzero. Thus, if q > q > e +1 ), then this polynomial must be thezero polynomial and hence αη /η = α σ η /η σ − α σ η = 0, which implies α = 0, a contradiction.Assume q = 3. Then σ = id, and αη η ℓ + (cid:16) αη η − αη (cid:17) ℓ + α η η − β = 0 holds for every ℓ ∈ F ,which implies αη /η + ( αη /η − αη ) = 0. Hence η = − αℓ + α ℓ/η + γ − αη /η = 0, which cannot hold for every ℓ ∈ F , a contradiction.Therefore, η = 0. Consequently, we have α = 0 by (10).If η = 0, then γ = 0 by the third equation of (10), and the second equation of (11) becomes( x σ +1 − ℓ σ +1 ) η = β , which has a solution x ∈ F q for every ℓ ∈ F q . By our assumption, β = 0.For ℓ = 0 there exists t ∈ F × q such that t σ +1 = β/η . For ℓ = t there exists z ∈ F q suchthat z σ +1 = t σ +1 + β/η = t σ +1 + t σ +1 = − t σ +1 . Thus ( z/t ) σ +1 = −
1. Let δ be a generatorof the cyclic group F × q and set z/t = δ n ( n > − δ n ) σ +1 = ( δ σ +1 ) n has order2 in F × q . Since | δ σ +1 | = ( q − /
2, we have 2 = | ( δ n ) σ +1 | = ( q − / (2 · gcd(( q − / , n )) andgcd(( q − / , n ) = ( q − /
4. But 4 ∤ ( q −
1) as q is an odd power of 3, a contradiction. Therefore η = 0, and the third equation of (11) becomes x σ +2 = γ/η + ℓ σ +2 . Since x ( x σ +2 ) σ − =( x σ − ) = 1 for x ∈ F × q , we have x = ( x σ +2 ) / ( x σ +2 ) σ = ( γ/η + ℓ σ +2 ) / ( γ/η + ℓ σ +2 ) σ when13 /η + ℓ σ +2 = 0, and x σ +1 ( γ/η + ℓ σ +2 ) = ( γ/η + ℓ σ +2 ) σ . The latter also holds when γ/η + ℓ σ +2 = 0. Multiplying ( γ/η + ℓ σ +2 ) on both sides of the second equation of (11), we obtain βℓ σ +2 + γη η ℓ σ +1 + βγη − γ σ η η σ = 0 , (12)which holds for every ℓ ∈ F q . Since the polynomial of ℓ on the left-hand side of this equationhas at most 3 e +1 + 2 roots if it is nonzero, we must have β = 0 and γη /η = 0 if q >
3. If q = 3, then σ = id and (12) becomes βℓ + γη η ℓ + βγη − γη η = 0, and so we still have β = 0 and γη /η = 0. By our assumption, γ = 0 and thus η = 0. Lemma 3.7.
Let G = h R( q ) , ζ i (as given in (7) ) be a subgroup of Sym( V ) containing R( q ) as anormal subgroup. Suppose that P is an imprimitive block of G ∞ on V \ {∞} containing suchthat P \ { } is an R( q ) ∞ , -orbit on V \ {∞ , } of length q − . Then the foollowing hold: (a) P = { (0 , , η, . . . ) ∈ V | η ∈ F q } and G ∞ ,P = h n κ , t , ,ξ , ζ | ξ ∈ F q , κ ∈ F × q i ; (b) setting L := P ∪ {∞} , D := ( V, L R( q ) ) = ( V, L G ) is a - ( q + 1 , q + 1 , q + 1) designadmitting G as a -point-transitive and block-transitive group of automorphisms, and Ω :=( ∞ , L ) R( q ) = ( ∞ , L ) G is a 1-feasible G -orbit on the flag set of D ; moreover, the G -flaggraph of D with respect to (Ω , Ξ) is the same as the R( q ) -flag graph Γ( D , Ω , Ξ) with respectto (Ω , Ξ) ; (c) Γ( D , Ω , Ξ) is connected with order | Ω | = q ( q + 1) and valency q when q > and has threeconnected components when q = 3 . Proof. (a) By Lemma 3.5 and Lemma 3.6, P = { (0 , , η, . . . ) ∈ V | η ∈ F q } . By (8) one cancheck that P is indeed an imprimitive block of G ∞ on V \ {∞} , and G ∞ ,P = h n κ , t , ,ξ , ζ | ξ ∈ F q , κ ∈ F × q i .(b) Since w does not stabilize L , we have λ = q + 1 for D .(c) We first recall the following known result (see [23, p.60, Theorem C] or [13, p.3758,Lemma 2.2]): For any subgroup H of R( q ), either | H | = ( s + 1) s ( s − s m = q forsome positive integer m , or | H | divides q ( q − q + 1), q − q , 6( q + √ q + 1), 6( q − √ q + 1),504 or 168.By Lemma 2.5, it suffices to prove R( q ) = h t , ,ξ , n κ , w | ξ ∈ F q , κ ∈ F × q i when q > ′ = h t , ,ξ , n − , w | ξ ∈ F i . Denote H := h t , ,ξ , n κ , w | ξ ∈ F q , κ ∈ F × q i . Since for ξ, η ∈ F × q and θ ∈ F q , t , ,ξ (0 , , ξ, , − ξ σ , − ξ ) w (cid:18) ξ σ − , , − ξ , , , − ξ (cid:19) t , ,θ (cid:18) ξ σ − , , θ − ξ , . . . (cid:19) , ( η, , , − η σ +3 , − η σ +3 , − η σ +4 ) w (cid:18) − η , − η σ +1 , , . . . (cid:19) t , ,θ (cid:18) − η , − η σ +1 , θ, . . . (cid:19) , (13)we see that { ( ζ, , η, . . . ) ∈ V | ζ, η ∈ F q } ∪ { ( ζ, − ζ σ +1 , θ, . . . ) ∈ V | ζ ∈ F × q , θ ∈ F q } is includedin the H -orbit containing ∞ and thus | H | = |∞ H || H ∞ | ≥ (2 q − q + 1) q ( q − q ≥
27, we have | H | = ( s + 1) s ( s −
1) by the above-mentioned result, where s m = q for some positive odd integer m . It follows that m = 1 and H = R( q ). When q = 3, we usethe permutation representation of R(3) as a primitive group of degree 28 in the database ofprimitive groups in Magma [5]. Now R(3) acts on ∆ := { , , . . . , } , and the two actions ofR(3) on V and ∆ are permutation isomorphic. Let J be the normal subgroup of R(3) (thestabilizer of 1 ∈ ∆ in R(3)) which is regular on ∆ \ { } , and let Z be the centre of J . Then H is (permutation) isomorphic to h Z, R(3) , , τ i for some involution τ ∈ R(3). Computation in
Magma shows that h Z, R(3) , , τ i has order 6 ,
18 or 504 for any involution τ of R(3). Hence14 H | = 504 as | H | ≥ · · ′ is the only subgroup of R(3) of order 504, wehave H = R(3) ′ . Case (iii): | P | = q , L ∪ L ⊆ P Now { (0 , η, , . . . ) | η ∈ F q } ⊆ P . Since (0 , η, , . . . ) t α,β,γ = ( α, η + β, γ − αη, . . . ), we have h t ,β, | β ∈ F q i ≤ G ∞ ,P and H := h t ,β, , n κ | β ∈ F q , κ ∈ F × q i ≤ G ∞ ,P . Lemma 3.8.
Let G = h R( q ) , ζ i (as given in (7) ) be a subgroup of Sym( V ) containing R( q ) as anormal subgroup. Suppose that P is an imprimitive block of G ∞ on V \ {∞} containing suchthat | P | = q and L ∪ L ⊆ P . Then the following hold: (a) P = { (0 , η , η , . . . ) | η , η ∈ F q } and G ∞ ,P = h n κ , t ,ξ,η , ζ | ξ, η ∈ F q , κ ∈ F × q i ; (b) setting L := P ∪ {∞} , D := ( V, L R( q ) ) = ( V, L G ) is a - ( q + 1 , q + 1 , q + 1) designadmitting G as a -point-transitive and block-transitive group of automorphisms, and Ω :=( ∞ , L ) R( q ) = ( ∞ , L ) G is a 1-feasible G -orbit on the flag set of D ; moreover, the G -flaggraph of D with respect to (Ω , Ξ) is the same as the R( q ) -flag graph Γ( D , Ω , Ξ) with respectto (Ω , Ξ) ; (c) Γ( D , Ω , Ξ) is connected with order | Ω | = q ( q + 1) and valency q . Proof. (a) Since | P | = q by our assumption, it suffices to prove η = 0 for every ( η , η , η , . . . ) ∈ P . Suppose otherwise. By the action of H we may assume (1 , , ε , . . . ) =: ρ ∈ P for some ε ∈ F q . Since | ρ H | = | H | / | H ρ | = | H | = q ( q −
1) and ρ H ∩ { (0 , η, , . . . ) | η ∈ F q } = ∅ , we have P = ρ H ∪ { (0 , η, , . . . ) | η ∈ F q } = { ( η , η , η , . . . ) ∈ V | η = η σ +21 ε + η η } . So (0 , , , . . . ) and(1 , , ε , . . . ) are points in P . If (0 , , , . . . ) t x,y,z = (1 , , ε , . . . ), then x = 1 , y = − , z = 1 + ε .On the other hand, (0 , − , , . . . ) ∈ P , and (0 , − , , . . . ) t , − , ε = (1 , − , ε , . . . ) = (1 , , ε , . . . ) / ∈ P . Therefore, P is not an imprimitive block of R( q ) ∞ on V \ {∞} , a contradiction.(b) It is clear that D is a 2-( q + 1 , q + 1 , λ ) design admitting G as a 2-point-transitive andblock-transitive group of automorphisms, and Ω is a 1-feasible G -orbit on the flag set of D . If λ = 1, then R( q ) L is 2-transitive on L and thus | R( q ) L | = | L | · | R( q ) ∞ ,L | = ( q + 1) q ( q − | R( q ) L | should divide | R( q ) | = ( q + 1) q ( q − λ = q + 1.(c) By Lemma 2.5, it suffices to prove R( q ) = H := h t ,ξ,η , w | ξ, η ∈ F q i . In fact, similar to(13), we can see that H is transitive on V , and thus | H | = | V || H ∞ | is divisible by ( q + 1) q .Similar to the proof of part (c) of Lemma 3.7, when q ≥
27, we have | H | = ( s + 1) s ( s − s m = q for some odd positive integer m . It follows that m = 1 and H = R( q ). When q = 3, the action of R(3) on V is permutation isomorphic to the primitive action of R(3) on∆ := { , , . . . , } (see Magma [5]). Let J be the normal subgroup of R(3) which is regularon ∆ \ { } . J has two subgroups of order 9 which are normal in R(3) . One of them, say X , iselementary abelian, while the other is cyclic. So H is (permutation) isomorphic to e H := h X, τ i for some involution τ ∈ R(3). Computation in
Magma shows that | e H | = 18 or 1512 for anyinvolution τ in R(3). Since | H | ≥ ·
9, it follows that H = R(3). Case (iv): | P | = q , ( L ∪ L ) ∩ P = ∅ Lemma 3.9.
Let soc( G ) = R( q ) or G = R(3) . Suppose that P is an imprimitive block of G ∞ on V \ {∞} containing such that | P | = q and ( L ∪ L ) ∩ P = ∅ . Then the following hold: (a) G = R(3) , P = { ( a, − a , c, . . . ) ∈ V | a, c ∈ F } , and G ∞ ,P = h n − , t x, − x ,z | x, z ∈ F i ; (b) setting L := P ∪ {∞} , D := ( V, L
R(3) ) is a - (28 , , design admitting R(3) as a -point-transitive and block-transitive group of automorphisms, and Ω := ( ∞ , L ) R(3) is a1-feasible
R(3) -orbit on the flag set of D ; the R(3) -flag graph Γ( D , Ω , Ξ) has three connected components. Proof. (a) Since (0 , , , . . . ) t α,β,γ = ( α, β, γ, . . . ), if ( α, β, γ, . . . ) ∈ P , then t α,β,γ stabilizes P . Let( η , η , η , . . . ) ∈ P be a fixed element with η = 0. Since N := h n κ | κ ∈ F × q i stabilizes P , we mayassume that η = 1. Then (0 , , γ, . . . ) ∈ P for any γ ∈ F × q as t ,η ,η = t , , − and N stabilizes P . Thus, if β = 0, then (0 , β, γ, . . . ) / ∈ P , for otherwise (0 , β, , . . . ) = (0 , β, γ, . . . ) t , , − γ ∈ P ,which contradicts the assumption ( L ∪ L ) ∩ P = ∅ .Let (1 , ξ , ξ , . . . ) and (1 , θ , θ , . . . ) be two points in P . Then (1 , ξ , ξ , . . . ) N = (1 , θ , θ , . . . ) N holds if and only if ξ = θ and ξ = θ . Moreover, ( − , ξ , − ξ , . . . ) ∈ P and t − ,ξ , − ξ sta-bilizes P . Hence (1 , η , η , . . . ) t − ,ξ , − ξ = (0 , η + ξ − , η − ξ + η + ξ − , . . . ) ∈ P and so η + ξ − ξ = η , we obtain η = −
1. Therefore, P = { (0 , , γ, . . . ) | γ ∈ F q } ∪ (cid:0) ∪ c ∈ F q (1 , − , c, . . . ) N (cid:1) = { ( a, − a σ +1 , c, . . . ) ∈ V | a, c ∈ F q } . On the other hand,since ( a, − a σ +1 , c, . . . ) t x,y,z = ( a + x, − a σ +1 + y + x σ a, c + z + xa σ +1 + ya − x σ +1 a, . . . ), if( a , − a σ +10 , c, . . . ) t x,y,z ∈ P for some a ∈ F q , then y = x σ a − a σ x − x σ +1 . Thus, t x,y,z stabilizes P if and only if x σ a − a σ x − x σ +1 = x σ a − a σ x − x σ +1 for any a ∈ F q , that is, ( a − a ) σ x = ( a − a ) x σ for any a ∈ F q . Hence P = { ( a, − a σ +1 , c, . . . ) ∈ V | a, c ∈ F q } is an imprimitive block of R( q ) ∞ on V \ {∞} if and only if q = 3.(b) D is a 2-(28 , , λ ) design for some λ ≥
1. If λ = 1, then R(3) L is 2-transitive on L and thus | R(3) L | = | L | · | R(3) ∞ ,L | . But | R(3) L | should divide | R(3) | = 28 ·
54, a contradiction.Therefore, by Lemma 2.3, λ = 10.(c) Set H := h n − , t x, − x ,z , w | x, z ∈ F i . Then | H | = |∞ H || H ∞ | ≥ · · { , , . . . , } ,let J be the normal subgroup of R(3) which is regular on ∆ \ { } . Then H is (permutation)isomorphic to e H := h Y, R(3) , , τ i for some involution τ ∈ R(3), where Y is the cyclic subgroupof J of order 9 which is normal in R(3) . Using Magma , we find that | e H | = 18 or 504 for anyinvolution τ of R(3). Therefore, H = R(3) ′ and Γ( D , Ω , Ξ) has three connected components byLemma 2.5.
In this section we assume that G is a 2-transitive permutation group acting on a set V whichwe always assume to be a suitable vector space over some finite field, and soc( G ) is abelian. Let u := | V | = p d be the degree of this permutation representation, where p is a prime and d ≥ u and the stabilizer G of the zero vector in G are as follows.(i) G ≤ ΓL(1 , q ), q = p d ;(ii) G ☎ SL( n, q ), n ≥ q n = p d ;(iii) G ☎ Sp( n, q ), n ≥ n is even, q n = p d ;(iv) G ☎ G ( q ), q = p d , q > q is even;(v) G = G (2) ′ ∼ = PSU(3 , u = 2 ;(vi) G ∼ = A or A , u = 2 ;(vii) G ∼ = SL(2 , u = 3 ;(viii) G ☎ SL(2 ,
5) or G ☎ SL(2 , d = 2, p = 5 , , , , ,
29, or 59;(ix) d = 4, p = 3. G ☎ SL(2 ,
5) or G ☎ E , where E is an extraspecial group of order 32.Since G is 2-transitive on V , in each case G is transitive on V \ { } . Denote by T thesubgroup of Sym( V ) consisting of all translations of V , so that G = T ⋊ G .16 .1 G ≤ ΓL(1 , q ) , q = p d This case gives (f) in Table 2. Now G acts on V = F q , and a typical element of G is of the form τ ( a, c, ϕ ) : F q → F q , z az ϕ + c, where a ∈ F × q , c ∈ F q and ϕ ∈ Aut( F q ) = h ζ i , with ζ : F q → F q , z z p the Frobenius map. Denote t ( a, j ) := τ ( a, , ζ j ) , where j is an integer. Given δ ∈ Aut( F q ) and integer i ≥
0, we use [ δ, i ] to denote ( p ni − / ( p n − δ − p n −
1, where n is the smallest positive integer such that δ = ζ n . Thus,for i > x ∈ F × q , x [ δ,i ] is the product of x δ i − , x δ i − , . . . , x δ , x in F × q . The following twolemmas can be easily proved. Lemma 4.1.
Let H be a subgroup of F × q . Let x ∈ F × q \ H and δ ∈ Aut( F q ) . In the sequence: H , Hx [ δ, , Hx [ δ, , . . . , Hx [ δ,n ] , . . . , if j is the smallest positive integer such that Hx [ δ,j ] equalsa previous item, then Hx [ δ,j ] = H . Lemma 4.2.
Let H be a subgroup of F × q and x ∈ F × q . Then x ∈ H if and only if x | H | = 1 . Let a and c be coprime positive integers. Denote by ord p ( a ) the exponent of a prime p in a , namely the largest nonnegative integer i such that p i | a . If j is the smallest positive integersuch that c | ( a j − a has order j (mod c ). Denote by S ( n ) the set of primedivisors of a positive integer n . Lemma 4.3.
Let m > and a > be coprime integers. (a) If m is odd, then a has order m (mod ( a − m ) if and only if S ( m ) ⊆ S ( a − ; (b) if m is even, then a has order m (mod ( a − m ) if and only if S ( m ) ⊆ S ( a − , andeither a ≡ or ∤ m . Proof.
Let p be a prime not dividing a and suppose a has order f (mod p ). Suppose that n is a positive integer and f | n . Then by [1, pp.355–356, 1), 2), 3), 4)], we have ord p ( a n −
1) =ord p ( a f − p ( n ) when p or n is odd, and ord ( a n −
1) = ord ( a − ( a +1)+ord ( n ) − p = 2 and n is even.First assume that a has order m (mod ( a − m ). Suppose S ( m ) * S ( a −
1) and let p bethe largest one in S ( m ) \ S ( a − p is odd and p ∤ a as gcd( a, m ) = 1 by our assumption.Suppose that a has order f (mod p ). Then f | m as p divides a m −
1. Set m = m/p . Let ξ ∈ S (( a − m ) = S ( a − ∪ S ( m ) and suppose that a has order f ξ ( mod ξ ). If ξ ∈ S ( m ) \ S ( a − ξ is odd, f ξ | m since f ξ < ξ ≤ p , and ord ξ ( a m −
1) = ord ξ ( a f ξ − ξ ( m ) ≥ ord ξ ( m ) =ord ξ (( a − m ). If ξ ∈ S ( a − \ S ( m ), then ord ξ ( a m − ≥ ord ξ ( a −
1) = ord ξ (( a − m ). If ξ ∈ S ( m ) ∩ S ( a − ξ ( a m − ≥ ord ξ ( a −
1) + ord ξ ( m ) = ord ξ ( a −
1) + ord ξ ( m ) =ord ξ (( a − m ). It follows that ( a − m | ( a m − S ( m ) ⊆ S ( a − m = 2 ℓ , we need to prove either a ≡ ∤ m . Suppose a ≡ − | m , then ord ξ ( a ℓ −
1) = ord ξ ( a −
1) + ord ξ ( ℓ ) = ord ξ ( a −
1) + ord ξ ( m ) when ξ ∈ S ( m ) isodd, andord ( a ℓ −
1) = ord ( a −
1) + ord ( a + 1) + ord ( ℓ ) − ≥ ord ( a −
1) + ord ( m ) , which implies ( a − m | ( a ℓ − S ( m ) ⊆ S ( a − m is even, we also assume 4 ∤ m or a ≡ a − m | ( a m − t be a positive integer such that ( a − m | ( a t − p ∈ S ( m ), we have ord p ( t ) ≥ ord p ( m ). Hence if m is odd, then m ≤ t .Suppose that m is even. Then t has to be even (otherwise ord ( a t −
1) = ord ( a −
1) and a t − a − m ). If 4 ∤ m , then obviously m ≤ t . If a ≡ ( a t −
1) = ord ( a −
1) + ord ( t ) ≥ ord (( a − m ) and thus ord ( t ) ≥ ord ( m ), which implies m ≤ t . Therefore, a has order m (mod ( a − m ).Next we prove some results on the structure of G and determine all possible imprimitiveblocks of G on F × q containing 1.Denote by s the smallest positive integer such that t ( a, s ) ∈ G for some a ∈ F × q . Then s must be a divisor of d and { ℓ > | t ( a, ℓ ) ∈ G for some a ∈ F × q } = { js | j = 1 , , . . . } ,since t ( a , j ) t ( a , j ) = t ( a a ζ j , j + j ) ∈ G and t ( a i , j i ) − = t ((1 /a i ) ζ − ji , − j i ) ∈ G for t ( a i , j i ) ∈ G , i = 1 ,
2. If s = d , then G ≤ GL(1 , q ) and so G = GL(1 , q ) as G is transitive on F × q . For any integer i , set A i := { t ( a, is ) | t ( a, is ) ∈ G } , H i := { a | t ( a, is ) ∈ G } . Then A is a normal cyclic subgroup of G , H := H is a cyclic subgroup of F × q , and A i = A j ifand only if d | ( i − j ) s . Since A i t ( x, s ) j ⊆ A i + j for any i, j and t ( x, s ) ∈ A , | A i | is a constantand thus A i t ( x, s ) j = A i + j . Setting ϕ := ζ s , we then have A i = A t ( x, s ) i , H i = Hx [ ϕ,i ] , i = 1 , , . . . , d/s − , and A d/s = A t ( x, s ) d/s = A and H d/s = Hx [ ϕ,d/s ] = H . Since G = A ∪ A ∪ · · · ∪ A d/s − (disjoint union)is transitive on F × q , we have F × q = H ∪ H ∪ H ∪ · · · ∪ H d/s − . Denote by m the smallest positive integer such that t (1 , ms ) ∈ G , . Then m ≤ d/s , G , = h t (1 , ms ) i and | G , | = d/ ( ms ). Let x ∈ H . Since 1 ∈ H m = Hx [ ϕ,m ] , we have Hx [ ϕ,m ] = H . In the case when m >
1, if Hx [ ϕ,j ] = H for some j < m , then H j = H and t (1 , js ) ∈ A j ⊆ G , which contradicts the definition of m . Hence by Lemma 4.1, in the sequence: H , Hx [ ϕ, , Hx [ ϕ, , . . . , Hx [ ϕ,m − , Hx [ ϕ,m ] , . . . , the first m items are pairwise distinct, and thesubsequent items repeat the previous ones. Since G is transitive on F × q , we have F × q = H ∪ Hx [ ϕ, ∪ · · · ∪ Hx [ ϕ,m − , | F × q : H | = m, m | [ ϕ, m ] . (14) Lemma 4.4.
Let G ≤ AΓL(1 , q ) act -transitively on F q , where q = p d with p a prime, and let s and m be as above. (a) If m = 1 , then G is the group generated by GL(1 , q ) and τ (1 , , ζ s ) . Conversely, if G contains GL(1 , q ) , then m = 1 ; (b) if m > , then p s has order m (mod m ( p s − , S ( m ) ⊆ S ( p s − , and G has φ ( m ) possibilities, where φ ( m ) is the number of positive integers less than m and coprime to m . Proof.
Let A i , H i and H be defined as above, and ϕ := ζ s .(a) If m = 1, then H = F × q and G is generated by GL(1 , q ) and τ (1 , , ζ s ). Conversely, if G contains GL(1 , q ), then m = 1 by the definition of m .18b) Assume m >
1. Since G is transitive on F × q , we have Hx = H , Hx [ ϕ, = H , . . . , Hx [ ϕ,m − = H by (14) and Lemma 4.1, where x ∈ H . By Lemma 4.2, this is equivalent tosaying | H | = ( q − /m and x | H | = 1, x [ ϕ, | H | = 1, . . . , x [ ϕ,m − | H | = 1. Denote the set ofsolutions in F × q of each of the equations α | H | = 1 , α [ ϕ, | H | = 1 , . . . , α [ ϕ,m − | H | = 1by E , E , . . . , E m − , respectively. Then E i ( i = 1 , , . . . , m −
1) is a cyclic subgroup of F × q with | E i | = gcd( q − , [ ϕ, i ] | H | ) = | H | · gcd( m, [ ϕ, i ]), and E i /H is a subgroup of F × q /H of ordergcd( m, [ ϕ, i ]). Hence the existence of x satisfying (14) implies that m | ( q −
1) and F × q = ∪ m − i =1 E i (that is, ρ / ∈ ∪ m − i =1 E i , where ρ is a generator of F × q ), or equivalently m ∤ ( p si − / ( p s − i = 1 , , . . . , m −
1, and m | ( p sm − / ( p s − p s has order m (mod m ( p s − S ( m ) ⊆ S ( p s −
1) by Lemma 4.3.Moreover, by (14), { [ ϕ, , . . . , [ ϕ, m ] } is a complete residue system modulo m . It follows that ∪ m − i =1 ( E i /H ) is the set of all non-generators of F × q /H . Let ξ be a fixed generator of F × q . Then F × q \ ∪ m − i =1 E i = ∪ φ ( m ) i =1 Hξ ℓ i , where { ℓ = 1 , ℓ , . . . , ℓ φ ( m ) } is a reduced residue system modulo m ,and G is the group generated by { t ( a, | a ∈ H } and t ( ξ ℓ i , s ) for some i ∈ { , , . . . , φ ( m ) } . Lemma 4.5.
Let G ≤ AΓL(1 , q ) act -transitively on F q , where q = p d with p a prime. Let s, m, ϕ, H, A i and H i be as above. A subset P of F × q is an imprimitive block of G on F × q containing if and only if it is of the form P = K ∪ Kw [ ψ, ∪ Kw [ ψ, ∪ · · · ∪ Kw [ ψ,m/e − , (15) where e is a divisor of m , ψ := ζ es , and K is a subgroup of H such that w [ ψ,m/e ] ∈ K for some t ( w, es ) ∈ A e . Proof.
Suppose that P is an imprimitive block of G on F × q containing 1. Then G , ≤ G ,P ≤ G ≤ ΓL(1 , q ). Denote by e the smallest positive integer such that t ( c, es ) ∈ G ,P for some c ∈ F × q . Then G ,P ⊆ A ∪ A e ∪ A e ∪ A e ∪ · · · . For any integer i , set C i := { t ( a, ies ) | t ( a, ies ) ∈ G ,P } , K i := { a | t ( a, ies ) ∈ G ,P } . Then K := K is a subgroup of H . Set ψ := ϕ e = ζ es . For any t ( w, es ) ∈ G ,P , we have A je = A t ( w, es ) j , H je = Hw [ ψ,j ] , C j = C t ( w, es ) j , K j = Kw [ ψ,j ] , j = 1 , , . . . , m/e − . (16)Let ℓ be the smallest positive integer such that Kw [ ψ,ℓ ] = K . Then t (1 , ℓes ) ∈ G , . Since G , ≤ G ,P , by the definition of m , we have m = eℓ and so e divides m . By Lemma 4.1, in thesequence: K , Kw [ ψ, , Kw [ ψ, , . . . , Kw [ ψ,m/e − , Kw [ ψ,m/e ] , . . . , the first m/e items are pairwisedistinct, and the subsequent items repeat the previous ones. Since G ,P is transitive on P , P must be of the form (15) and moreover Kw [ ψ,m/e ] = K .We now prove that any subset P of F × q defined in (15) is an imprimitive block of G on F × q . Let e, ψ and K be as stated in the lemma. By Lemma 4.1 and the definition of m , in the sequence: K , Kw [ ψ, , Kw [ ψ, , . . . , Kw [ ψ,m/e − , Kw [ ψ,m/e ] , . . . , the first m/e terms must be pairwise distinct,and the subsequent terms repeat the previous ones as w [ ψ,m/e ] ∈ K . For any t ( z, ns ) ∈ G ( n > e ∤ n , then K , Kw [ ψ, , Kw [ ψ, , . . . , Kw [ ψ,m/e − are mapped into H n , H e + n , H e + n , . . . , H m − e + n respectively by t ( z, ns ), and hence P ∩ P t ( z,ns ) = ∅ . If e | n , say n = ej , then Kw [ ψ,i ] is mapped into H e ( i + j ) by t ( z, ns ). Since t ( z, ns ) ∈ A n = A t ( w, es ) j , we have z ∈ H n = Hw [ ψ,j ] and z = yw [ ψ,j ] for some y ∈ H , and ( Kw [ ψ,i ] ) t ( z,ns ) = yw [ ψ,j ] K ( w [ ψ,i ] ) ψ j = Kyw [ ψ,i + j ] . Hence( Kw [ ψ,i ] ) t ( z,ns ) = Kw [ ψ,i + j ] if y ∈ K , and ( Kw [ ψ,i ] ) t ( z,ns ) ∩ Kw [ ψ,i ] = ∅ if y / ∈ K . Therefore, y ∈ K implies P t ( z,ns ) = P , and y / ∈ K implies P ∩ P t ( z,ns ) = ∅ . Thus P is an imprimitive blockof G on F × q . 19 emma 4.6. Let
G, s, m, ϕ, A i , H i and H be as in Lemma 4.5, and by the proof of Lemma4.4 we may assume H = Hξ ℓ , where ξ is a fixed generator of F × q and ℓ is a positive integercoprime to m . Then a subset P of F × q containing is an imprimitive block of G on F × q if andonly if either P is a subgroup of F × q , or there exist j = nm + ℓ [ ϕ, e ] and K ≤ H with | H/K | dividing j [ ψ, m/e ] /m such that P is given by (15) , where n ≥ , e is a divisor of m , ψ := ζ es and w := ξ j . Proof.
By Lemma 4.5, we may assume P is given by (15). If m = 1, then e = m = 1, w = w [ ψ,m/e ] ∈ K , and thus P = K is a subgroup of F × q .Assume m > K of H in Lemma 4.5 must allow theexistence of w ∈ H e such that w [ ψ,m/e ] ∈ K . We may assume w = ξ nm + ℓ [ ϕ,e ] for some n ≥ H e = Hξ ℓ [ ϕ,e ] and H = h ξ m i . Set j = nm + ℓ [ ϕ, e ]. By Lemma 4.3, e | [ ϕ, e ] and m/e divides[ ψ, m/e ]. Thus e | j and m | j [ ψ, m/e ], and w [ ψ,m/e ] ∈ K ⇔ ξ j [ ψ,m/e ] | K | = 1 ⇔ ( q − | j [ ψ, m/e ] | K | ⇔ | H || K | | j [ ψ, m/e ] m . (17)Thus P in (15) is an imprimitive block of G on F × q if and only if K is a subgroup of H suchthat | H/K | divides j [ ψ, m/e ] /m . Remark 4.7.
We show that P in Lemma 4.5 needs not be a subgroup of F × q . Let ξ, ℓ, j and w be as in Lemma 4.6. Let K ≤ H be such that | H/K | divides j [ ψ, m/e ] /m . Then P givenby (15) is an imprimitive block of G on F × q . Moreover, P is a subgroup of F × q if and only if( Kw ) m/e = K , that is, if and only if | H/K | divides j/e .Fix s, m and e . Let µ be an odd prime divisor of [ ψ, m/e ] / ( m/e ), and let t := ord µ ( j/e ) ≥ p has order f (mod µ ) so that ord µ ( p d −
1) = ord µ ( p f −
1) + ord µ ( d ). Choose d such that ord µ (( p d − /m ) ≥ t + 1. Let K be the subgroup of H of index µ t +1 in H . Now µ t +1 | j [ ψ, m/e ] /m , while j/e cannot be divided by µ t +1 . So P is an imprimitive block of G on F × q , but not a subgroup of F × q .In the rest of this section, let G, H, K, P, m, s, ϕ, e be as in Lemma 4.5. Set L := P ∪ { } and D := ( F q , L G ) , Ω := (0 , L ) G . Then D is a 2-( q, | P | + 1 , λ ) design admitting G as a 2-point-transitive and block-transitivegroup of automorphisms, and Ω is a 1-feasible G -orbit on the flag set of D . Next we considerthe connectedness of the G -flag graphs.If λ = 1, then by [22, Proposition 4.1], L is a subfield of F q . Conversely, if L is a subfieldof F q , then the elements in G interchanging 0 and 1 must stabilize L and thus λ = 1. In thecase when L is a subfield of F q with | L | = p t , we have Γ( D , Ω , Ξ) ∼ = ( p d − p d / (( p t − p t ) · K p t ,yielding the first possibility in (f) in Table 2.In what follows we assume that L is not a subfield of F q so that λ >
1. Denote by e T and e T the subgroups of the addition group ( F q , +) generated by { a − | a ∈ P } and { a | a ∈ P } ,respectively. Define T := { τ (1 , c, id) | c ∈ e T } , T := { τ (1 , c, id) | c ∈ e T } . Since | H | = ( q − /m = q − ϕ m − ϕ,m ] m ( ϕ −
1) is even when p is odd, we have τ ( − , , id) ∈ G and τ ( − , , id) ∈ G . Set − P := {− a | a ∈ P } . We observe that for p > P = − P (or equivalently τ ( − , , id) stabilizes P ) if and only if | K | is even. Lemma 4.8.
Let J := h G ,P , τ ( − , , id) i . Then J = ( T ⋊ G ,P ) ⋊ h τ ( − , , id) i if P = − P ,and J = T ⋊ G ,P if P = − P . roof. Let κ := τ ( − , , id). It is not difficult to verify that G ,P normalizes T i , i = 1 ,
2. If σ = τ ( a, , θ ) ∈ G ,P , then κσ − κσ = τ (1 , a − , id). Hence T ≤ J .Every element σ κσ · · · σ n − κσ n of J is in J or J κ , where J := T ⋊ G ,P and σ i ∈ G ,P , i = 1 , , . . . , n . We can see that κ normalizes J , and when P = − P we have κ / ∈ J as τ ( − , , id) / ∈ G ,P . When P = − P , we have τ ( − , , id) ∈ G ,P and thus T ≤ J . Lemma 4.9.
Suppose that < | P | < q − and | T i | = p c i , i = 1 , . Then Γ( D , Ω , Ξ) has | H/K | ep d − c / connected components when P = − P , and | H/K | ep d − c connected componentswhen P = − P . Therefore, Γ( D , Ω , Ξ) is connected if and only if p ≡ − , d is odd,and P is the subgroup of F × q of index . Moreover, if Γ( D , Ω , Ξ) is connected, then it has order | Ω | = 2 q and valency ( q − / . Proof.
By Lemma 2.5, the number of connected components of Γ( D , Ω , Ξ) is equal to | G : J | = | G | − i p c i | G ,P | = | G : G ,P | − i p c i = p d ( p d − − i p c i | K | m/e = | H/K | ep d − c i − i , where i = 1 if P = − P and i = 2 if P = − P .If P = − P , then Γ( D , Ω , Ξ) must be disconnected, for otherwise we would have c = d , K = H , e = 1 and so P = F × q , a contradiction.Assume that P = − P and Γ( D , Ω , Ξ) is connected. Then p > | K | is odd, c = d and | H/K | · e = 2. If | H/K | = 1 and e = 2, then | K | = | H | = ( q − /m = q − ϕ m − ϕ,m ] m ( ϕ −
1) is even,a contradiction. So we have | H/K | = 2 and e = 1. By Lemma 4.6, j [ ϕ, m ] /m is even, where j = nm + ℓ for some n ≥ ℓ coprime to m . Since ord ( q −
1) = ord ( | H | ) + ord ( m ) =1 + ord ( m ) and q − q − ϕ m − [ ϕ, m ]( ϕ − ∤ [ ϕ, m ] /m , 2 | j and m is odd asgcd( m, ℓ ) = 1. It follows that ord ( q −
1) = 1, p ≡ − d is odd. By Remark 4.7, P is a subgroup of F × q with index ( q − / ( | K | m/e ) = 2.Conversely, if p ≡ − d is odd and P is the subgroup of F × q of index 2, then | P | is odd and P = − P . If e T = F q , then | F q : e T | = q/ | e T | ≥ p . Hence 2 = ( q − / | P | =( q − / | P − | ≥ ( q − / | e T | ≥ p (1 − /q ) as P − ⊆ e T , which implies p = 3, d = 1 and thus | P | = 1. Therefore, if | P | >
1, then e T = F q and consequently Γ( D , Ω , Ξ) is connected. G ☎ Sp( n, q ) , n ≥ , n is even, u = q n = p d This case contributes to (g) in Table 2. Denote the underlying symplectic space by (
V, f ), where V = F nq and f is a symplectic form. Let e := (1 , , . . . ,
0) and H := Sp( n, q ) ✂ G . Define C α := { z ∈ V \ h e i | f ( z , e ) = α } , α ∈ F q . By Witt’s Lemma each C α is an H e -orbit on V \ h e i . Moreover, | C | = q n − − q and | C α | = q n − for α ∈ F × q .A typical element of G ≤ AΓL( n, q ) is τ ( A, c , ϕ ) : x x ϕ A + c , where A ∈ GL( n, q ), c ∈ V and ϕ is a field automorphism of F q acting componentwise on x . If τ ( A, c , ϕ ) ∈ G h e i , then c ∈ h e i and row ( A ) ∈ h e i , where row ( A ) is the first row of A . With respect to the bijection ρ : h e i → F q , a e a , each τ ( A, c , ϕ ) ∈ G h e i induces a permutation e τ ( a, c, ϕ ) : F q → F q , z az ϕ + c , where a e = row ( A ) and c e = c . Define η : G h e i → AΓL(1 , q ), τ ( A, c , ϕ ) e τ ( a, c, ϕ ). Then η is a homomorphism. Lemma 4.10.
Suppose that P is a nontrivial imprimitive block of G on V \ { } containing e . Then (a) P ⊆ h e i and ρ ( P ) is a subgroup of F × q ; (b) D := ( V, L G ) is a - ( q n , | P | + 1 , λ ) design admitting G as a group of automorphisms, where L := P ∪ { } , and Ω := ( , L ) G is a 1-feasible G -orbit on the flag set of D ; moreover, λ = 1 if and only if ρ ( P ) ∪ { } is a subfield of F q ; D , Ω , Ξ) is disconnected. Proof.
We first prove that C * P . Suppose otherwise. Suppose that P includes j − H e of length q n − (1 ≤ j < q + 1) and contains ℓ elements in h e i (1 ≤ ℓ < q ). Then | P | = jq n − + ℓ − q = gcd( jq n − + ℓ − q, q n −
1) = gcd( q n − , q − ℓq − j ). If q − ℓq − j = 0,then jq n − + ℓ − q ≤ q − ℓq − j , which is impossible as n ≥
4. If q − ℓq − j = 0, then j = q , ℓ = q −
1, and thus P = V \ { } , violating the condition ( u − / | P | ≥
2. Therefore, C * P .Suppose that P includes j orbits of H e of length q n − (0 ≤ j < q ) and contains ℓ elementsin h e i (1 ≤ ℓ < q ). Then | P | = jq n − + ℓ = gcd( jq n − + ℓ, q n −
1) = gcd( q n − , ℓq + j ) ≤ ℓq + j .Since n ≥
4, it follows that j = 0 and P ⊆ h e i . Since P is an imprimitive block of G h e i , on h e i \ { } , ρ ( P ) is an imprimitive block of η ( G h e i , ) on F × q containing 1.Let e = a , b , a , b , . . . , a t , b t be a symplectic basis of ( V, f ), where ( a i , b i ) is a hyperbolicpair, i = 1 , , . . . , t . With respect to this basis, for any a ∈ F × q , we have τ ( A ( a ) , , id) ∈ Sp( n, q ) ∩ G h e i , , where A ( a ) := diag( a, /a, , , . . . , , η ( G h e i , ) contains GL(1 , q ) and by thediscussion in Section 4.1, ρ ( P ) = { a ∈ F × q | a e ∈ P } is a subgroup of the multiplicative group F × q .Since P and h e i \ { } are both imprimitive blocks of G on V \ { } , we have G ,P ≤ G , h e i .Let g be an element of G h e i interchanging and e . Then J := h G ,P , g i ≤ G h e i = G , andthus the G -flag graph Γ( D , Ω , Ξ) is disconnected by Lemma 2.5.The λ of D is 1 or | P | + 1 by Lemma 2.3. Since η ( G h e i ) is 2-transitive on F q , the argumentin Section 4.1 shows that λ = 1 if and only if ρ ( P ) ∪ { } is a subfield of F q . G ☎ SL(2 , q ) = Sp(2 , q ) , u = q = p d This case contributes to the second possibility in (g) in Table 2. Denote the underlying sym-plectic space by (
V, f ), where V = F q and f is a symplectic form. Let e := (1 ,
0) and H := Sp(2 , q ) = SL(2 , q ) ✂ G . Define C α as in Section 4.2. Then C = ∅ and C α = h e i + z α for α ∈ F × q , where z α ∈ C α . Denote all 1-subspaces of V by U = h e i , U , . . . , U q . Lemma 4.11.
Suppose that P is a nontrivial imprimitive block of G on V \ { } containing e . Then (a) P ⊆ h e i and { a ∈ F × q | a e ∈ P } is a subgroup of F × q ; (b) D := ( V, L G ) is a - ( q , | P | + 1 , λ ) design admitting G as a group of automorphisms, where L := P ∪ { } , and Ω := ( , L ) G is a 1-feasible G -orbit on the flag set of D ; moreover, λ = 1 if and only if { a ∈ F × q | a e ∈ P } ∪ { } is a subfield of F q ; (c) Γ( D , Ω , Ξ) is disconnected. Proof.
Suppose that P * h e i . Write P = ( U + z α ) ∪ · · · ∪ ( U + z α t ) ∪ E (1 ≤ t < q ),where E is a subset of h e i of size ℓ (1 ≤ ℓ < q ), α , . . . , α t are distinct elements of F × q and z α j ∈ C α j for 1 ≤ j ≤ t . Since H is transitive on the set of 1-subspaces of V , there exists γ ∈ H such that U γ = U . Hence P γ = ( U + z γα ) ∪ · · · ∪ ( U + z γα t ) ∪ E γ . Since U and U are not parallel, P γ ∩ P = ∅ and thus P = P γ ⊇ U + z γα . Since | ( U + z γα ) ∩ U | = 1 and | ( U + z γα ) ∩ ( U + z α j ) | = 1 for 1 ≤ j ≤ t , we have t + 1 ≥ | U + z γα | = q and thus t = q − | P | = q − q + ℓ = gcd( q − q + ℓ, q −
1) = gcd( q − , q − ℓ − ℓ = q − P = V \ { } , violating the condition ( u − / | P | ≥
2. Therefore, P ⊆ h e i .Similar to the treatment in Section 4.2, one can show that the set { a ∈ F × q | a e ∈ P } is asubgroup of F × q . Moreover, the G -flag graph Γ( D , Ω , Ξ) is disconnected, and λ = 1 if and onlyif { a ∈ F × q | a e ∈ P } ∪ { } is a subfield of F q . 22 .4 G ☎ SL( n, q ) , n ≥ , u = q n = p d This case contributes to the second possibility in (g) in Table 2. Let P be a nontrivial imprimitiveblock of G on V \{ } containing e := (1 , , . . . , V = F nq . Since V \h e i is a G , e -orbitof length q n − q , if P includes this orbit, then P = V \ { } as | P | is a divisor of | V \ { }| = q n − u − / | P | ≥
2. If P does not include V \ h e i , then P ⊆ h e i and similar to the argument in Section 4.2 one can show that { a ∈ F × q | a e ∈ P } is a subgroupof the multiplicative group F × q . Set L := P ∪ { } , D := ( V, L G ) and Ω := ( , L ) G . Then D is a2-( q n , | P | + 1 , λ ) design admitting G as a group of automorphisms in its natural action, where λ = 1 or | P | + 1, and Ω is a 1-feasible G -orbit on the flag set of D . Moreover, the G -flag graphΓ( D , Ω , Ξ) is disconnected, and λ = 1 if and only if { a ∈ F × q | a e ∈ P } ∪ { } is a subfield of F q . G ☎ G ( q ) , u = q = p d , q > , q is even This case contributes to the third possibility in (g) in Table 2. Suppose that P is a nontrivialimprimitive block of G on V \ { } containing e := (1 , , , , , V = F q . Then P is also an imprimitive block of G ( q ) on V \ { } and P is the union of some G ( q ) e -orbits on V \ { } . We are going to determine all possible orbit-lengths of G ( q ) e on V \ { } by using theknowledge of G ( q ) from [29, p.122, Section 4.3.4].Take a basis { x , x , . . . , x } of the octonion algebra O over F q with multiplication given byTable 3, or equivalently by Table 4, where e := x + x is the identity element of O (since thecharacteristic is 2, we omit the signs). x x x x x x x x x x x x x x x x x x x x x x x x x x x x x Table 3. Multiplication table of O e x x x x x x x e e x x x x x x x x x x e + x x x + x x x x x x x + x x x x x x x x x Table 4. Multiplication table of O There is a quadratic form N and an associated bilinear form f satisfying N ( x i ) = 0 and f ( x i , x j ) = ( , i + j = 9 , , i + j = 9 , i, j = 1 , , . . . , G ( q ) is the automorphism group of O . Since G ( q ) preserves the multiplicationtable of O , one can verify that it preserves N and f . Moreover, G ( q ) induces a faithful actionon e ⊥ / h e i , where e ⊥ = h x , x , x , x , x , x , e i . Hence G ( q ) can be embedded into Sp(6 , q ).There is also a symmetric trilinear form t on e ⊥ , that is, t ( z , z , z ) = t ( z σ , z σ , z σ ) for any σ ∈ Sym( { , , } ) and z , z , z ∈ e ⊥ (see [3, p.420] for the definition of a symmetric trilinearform), defined by t ( e , x , x ) = t ( e , x , x ) = t ( e , x , x ) = t ( x , x , x ) = t ( x , x , x ) = 1 (18)and otherwise t is zero on the basis vectors. It is straightforward to verify that t ( x , y , z ) = f ( xy , z ) , for x , y , z ∈ e ⊥ . (19)Since G ( q ) preserves the multiplication and f , it also preserves t .23enote by h x i the subspace of O spanned by x , and h x i the subspace of e ⊥ / h e i spanned by x , where x = x + h e i . If W is a subspace of O , then we use W to denote h W, e i / h e i . The actionsof G ( q ) on e ⊥ / h e i and V are permutation isomorphic.It is known that G ( q ) h x i has four orbits on the set of 1-subspaces of e ⊥ / h e i (see [9, Lemma3.1] and [22, p.72]), which have lengths 1, q , q ( q +1), q ( q +1) and are represented by h x i , h x i , h x i , h x i , respectively. (Since x is not perpendicular to x while x and x are perpendicularto x , h x i and h x i are in distinct G ( q ) h x i -orbits, and h x i and h x i are in distinct G ( q ) h x i -orbits. Moreover, h x i and h x i are also in distinct G ( q ) h x i -orbits. In fact, if this is not thecase, say ϕ ( h x i ) = h x i for some ϕ ∈ G ( q ) h x i , then ϕ ( x ) = a x + ℓ e , ϕ ( x ) = c x + s e forsome a, c = 0 and so = ϕ ( x ) ϕ ( x ) = ( a x + ℓ e )( c x + s e ) = ac x + ℓc x + as x + ℓs e , whichis a contradiction as ac = 0 and x , x , x , e are linearly independent.) Lemma 4.12. ([14, Lemma 4.8]) Let a ∈ V \ { } . Then G ( q ) a has q − orbits of length , q − orbits of length q , one orbit of length q ( q − and one orbit of length q ( q − on V \ { } . Lemma 4.13.
Let G ≤ AΓL(6 , q ) be 2-transitive on V such that G ☎ G ( q ) . Suppose that P is a nontrivial imprimitive block of G on V \ { } and let e ∈ P . Then the following hold: (a) P ⊆ h e i and { a ∈ F × q | a e ∈ P } is a subgroup of F × q ; (b) setting L := P ∪ { } , D := ( V, L G ) is a - ( q , | P | + 1 , λ ) design admitting G as a group ofautomorphisms, where λ = 1 or | P | + 1 , and Ω := ( , L ) G is a 1-feasible G -orbit on theflag set of D ; moreover, λ = 1 if and only if { a ∈ F × q | a e ∈ P } ∪ { } is a subfield of F q ; (c) Γ( D , Ω , Ξ) is disconnected. Proof.
Since P is the union of some G ( q ) e -orbits on V \ { } , we have a few combinations toconsider. We prove that only one possibility can actually occur.First, if P includes both the orbit of length q ( q −
1) and the orbit of length q ( q − P = V \ { } , contradicting theassumption that P is a nontrivial block of V \ { } .Next assume that P includes the orbit of length q ( q − i − q for some1 ≤ i < q +1 and ℓ orbits of length 1 for some 1 ≤ ℓ < q , but P does not include the orbit of length q ( q − | P | = iq − q + ℓ = gcd( iq − q + ℓ, q −
1) = gcd( iq − q + ℓ + ℓ ( q − , q −
1) =gcd( q ( ℓq + iq − , q −
1) = gcd( ℓq + iq − , q − < ℓq + iq −
1, we have iq − q + ℓ ≤ ℓq + iq − ≤ q −
1, which is impossible.Now assume that P includes the orbit of length q ( q − i orbits of length q for some0 ≤ i < q and ℓ orbits of length 1 for some 1 ≤ ℓ < q , but P does not include the orbit of length q ( q − | P | = iq + q − q + ℓ = gcd( iq + q − q + ℓ, q −
1) = gcd( iq + q − q + ℓ + ℓ ( q − , q − ℓq + iq + q − , q −
1) = gcd( ℓq + iq + q − , q − − ( ℓq + iq + q − ℓq + iq + q − , q − ℓq − iq − . (20)Since 0 < q − ≤ q − ℓq − iq − ≤ q − q −
1, we have iq + q − q + ℓ ≤ q − ℓq − iq − ≤ q − q −
1, which implies i = 0. Thus (20) gives | P | = q − q + ℓ = gcd( ℓq + q − , q − ℓq −
1) =gcd( q ( ℓq − q + ℓq +1) , q − ℓq −
1) = gcd( ℓq − q + ℓq +1 , q − ℓq −
1) = gcd( ℓq − q + ℓq +1 , q − q + ℓ ) = gcd( q − ℓq + ( ℓ − , q − q + ℓ ). Since 0 ≤ q − ℓq + ( ℓ −
1) = ( ℓ − q ) − ≤ q − q ,if q − ℓq + ( ℓ − = 0, then q − q + ℓ ≤ q − ℓq + ( ℓ − ≤ q − q , which is impossible.Hence q − ℓq + ( ℓ −
1) = 0, ℓ = q − | P | = q −
1. We claim that this P is not animprimitive block of G on V \ { } . As in [2, Section 1], for x ∈ e ⊥ , define x ∆ := { y ∈ e ⊥ | t ( x , y , z ) = 0 for any z ∈ e ⊥ } . ϕ ∈ G ( q ) x and suppose ϕ ( x ) = x + c e . Let y = a e + a x + a x + a x + a x + a x + a x be a vector in e ⊥ . If y ∈ ( x + c e ) ∆ , then t ( x + c e , y , z ) = 0 for any z ∈ e ⊥ . When z = e , x , x ,we get a = 0, a = 0, a = 0, and thus y ∈ W := h e , x , x , x i . It follows that ( x + c e ) ∆ ⊆ W .In addition, if c = 0, then x ∆ = h x , x , x i . Hence ϕ ( x ) ∈ ( x + c e ) ∆ ⊆ W , and ϕ ( x ) ∈ W ,which implies that the G ( q ) x -orbit containing x is included in W . Since by Lemma 4.12 thisorbit has length q ( q − W \ h x i , and P = W \ { } , where denotes the zerovector in e ⊥ / h e i .Suppose that W \ { } is an imprimitive block of G ( q ) on e ⊥ \ { } . Then there exists ψ ∈ G ( q ) stabilizing W such that ψ ( x ) = x . Let ψ ( x ) = x + j e for some j ∈ F q .Then 0 = t ( x , w , ψ − ( z )) = t ( ψ ( x ) , ψ ( w ) , z ) = t ( x + j e , ψ ( w ) , z ) for any w ∈ x ∆ and z ∈ e ⊥ . Hence ψ ( w ) ∈ ( x + j e ) ∆ and ψ ( x ∆ ) ⊆ ( x + j e ) ∆ . If j = 0, then ψ ( h x , x , x i ) ⊆ W ∩ h x , x , x i = h x , x i as x ∆ = h x , x , x i and ψ stabilizes W , a contradiction. If j = 0,then ( x + j e ) ∆ ⊆ h e , x , x , x i and ψ ( h x , x , x i ) ⊆ h e , x , x , x i∩h e , x , x , x i = h e , x , x i .Hence ψ ( h x , x , x i ) = h e , x , x i and e = ψ − ( e ) ∈ h x , x , x i , a contradiction. Therefore, W \ { } is not an imprimitive block of G ( q ) on e ⊥ \ { } .Therefore, the only possibility is that P includes neither the orbit of length q ( q −
1) northe orbit of length q ( q − C * P in Section 4.2, one can showthat P ⊆ h e i . Let e = c , c , . . . , c be a basis of V , and µ : e ⊥ / h e i → V a linear map thatmaps x , x , x , x , x , x to c , c , c , c , c , c , respectively. Up to permutation isomorphism,we only need to consider the action of G ( q ) on V defined by c g := µ (( µ − ( c )) g ) for c ∈ V and g ∈ G ( q ).Now G ≤ AΓL(6 , q ). Let ρ and η be defined as in Section 4.2. Similar to the proof ofLemma 4.12, one can verify that, for any a ∈ F × q , we have τ ( A ( a ) , , id) ∈ G ( q ) ∩ G h e i , ,where A ( a ) := diag(1 /a, a, , , /a, a ), with respect to the basis c , c , . . . , c . Hence η ( G h e i , )contains GL(1 , q ) and the discussion in Section 4.1 shows that ρ ( P ) = { a ∈ F × q | a e ∈ P } isa subgroup of F × q . Since G ,P ≤ G , h e i , J := h G ,P , g i ≤ G h e i = G , where g is an element in G h e i interchanging and e . Hence, by Lemma 2.5, the G -flag graph Γ( D , Ω , Ξ) is disconnected.The analysis in Section 4.1 shows that λ = 1 if and only if { a ∈ F × q | a e ∈ P } ∪ { } is a subfieldof F q . G = G (2) ′ ∼ = PSU(3 , , u = 2 Suppose that P is a nontrivial imprimitive block of G on V \ { } , where V = F . Let a ∈ P .Since 1 < | P | < | V | and the orbit-lengths of G , a on V \ h a i are 2(2 + 1), 2 (2 + 1), 2 and 2 (see [22, p.72]), P is the union of { a } and the G , a -orbit of length 2(2 + 1). Similar to the proofof Lemma 4.13, one can show that this P is not an imprimitive block of G (2) ′ on V \ { } . G ∼ = A or A , u = 2 Suppose that P is a nontrivial imprimitive block of G on V \ { } , where V = F . Let a ∈ P .Then G , a is transitive on V \ { , a } when G ∼ = A , and G , a has orbit-lengths 6 and 8 on V \ { , a } when G ∼ = A (see [22, p.72]). Hence there is no 2-design as in Lemma 2.4 admitting G as a group of automorphisms. If D is a 1-design with point set V admitting G as a group of automorphisms and Ω is a 1-feasible G -orbit on the set of flags of D , then the triple ( G, D , Ω) is said to be proper on V . Denote byΠ( V ) the set of all proper triples on V .The purpose of this subsection is to determine all proper triples in cases (vii)-(ix) in the open-ing paragraph of Section 4 up to flag-isomorphism to be defined as follows, and thus determine25ll possible flag graphs, contributing to (h), (i) and (j) in Table 2. Definition 4.14.
Let ( G i , D i , Ω i ) be a proper triple on V i , i = 1 ,
2. If there exists a bijection ρ : V → V such that the action of G on V is permutation isomorphic to the action of G on V with respect to ρ , and the action of G on Ω is permutation isomorphic to the actionof G on Ω with respect to the bijection from Ω to Ω induced by ρ , then ( G , D , Ω ) and( G , D , Ω ) are said to be flag-isomorphic with respect to the flag-isomorphism ρ .Flag-isomorphism is an equivalence relation on any subfamily Π of Π( V ). A subset of Π thathas exactly one proper triple from each equivalence class (of this equivalence relation) is calleda representative subset of Π.In what follows we set V := F dp , where d ≥ p is a prime, and denoteΠ( d, p ) := n ( G, D , Ω) ∈ Π( F dp ) | G ≤ AGL( d, p ) is 2-transitive on F dp o . Recall that T denotes the group of translations of V . Let ( G, D , Ω) , ( e G, e D , e Ω) ∈ Π( d, p ) be flag-isomorphic with respect to some ρ ∈ Sym( V ). Then ρ − Gρ = e G , ρ − T ρ = T and ρ − G ρ = e G .By [11, p.110, Corollary 4.2B], ρ ∈ N Sym( V ) ( T ) = AGL( d, p ). Since e G = ρ − G ρ = ( G ρ ) ρ = e G ρ and e G is transitive on V \ { } , we have ρ = and thus ρ ∈ GL( d, p ). Hence G and e G are conjugate in GL( d, p ), and up to flag-isomorphism it suffices to consider one representativein the conjugacy class of subgroups of GL( d, p ) containing G . We may thus assume G = e G in the following. Then ρ ∈ N GL( d,p ) ( G ). Since ρ is a flag-isomorphism, we have Ω ρ = e Ω, orequivalently ( , L ) ρ = ( , e L ) for some ( , L ) ∈ Ω and ( , e L ) ∈ e Ω as ρ normalizes G . Denote H := G ,L and e H := G , e L . Then H ρ = e H and so H and e H are conjugate in N GL( d,p ) ( G ). Upto flag-isomorphism it suffices to consider one representative in the conjugacy class of subgroupsof N GL( d,p ) ( G ) containing H . We may thus assume H = e H in the following. Lemma 4.15.
Let I be a finite group acting transitively on a set ∆ . Suppose that H ≤ J ≤ I such that J is transitive on ∆ , J x ≤ H and J y ≤ H for two points x, y in ∆ . Set P := x H and Q := y H . Then there exists σ ∈ Y := N I ( J ) such that ( P J ) σ = Q J if and only if x and y arein the same N Y ( H ) -orbit on ∆ . In particular, two systems of blocks P J and Q J are identical ifand only if x and y are in the same N J ( H ) -orbit on ∆ . Proof.
Assume that ( P J ) σ = Q J for some σ ∈ N I ( J ). Then y Hg = x Hσ and y = x τξ for some τ ∈ H and g ∈ J , where ξ := σg − . We need to show that ξ ∈ N Y ( H ). Set z := x τ = y ξ − .Then z H = x H = y Hgσ − = z ξHξ − , and we set M = z H . Since H z = ( H x ) τ = ( J x ) τ = J z =( J y ) ξ − = ( H y ) ξ − = ( ξHξ − ) z , we have H = J z H = J M = J z H ξ − = H ξ − .Conversely, if y = x σ for some σ ∈ N Y ( H ), then Q = y H = x Hσ = P σ and so ( P J ) σ = Q J .By Lemma 4.15, L \ { } = x H and e L \ { } = y H for some x, y that are in the same N Y ( H )-orbit on V , where Y = N GL( d,p ) ( G ). Therefore, we obtain the following result. Lemma 4.16.
Let V := F dp , where d ≥ and p is a prime. Suppose that GL( d, p ) ≤ Sym( V \{ } ) has exactly ℓ conjugacy classes C , . . . , C ℓ of transitive subgroups. For ≤ i ≤ ℓ , let G i ∈ C i and N i := N GL( d,p ) ( G i ) , and suppose that N i has exactly n ( i ) conjugacy classes H i, , . . . , H i,n ( i ) of subgroups of G i containing some point stabilizer of G i . For ≤ i ≤ ℓ and ≤ j ≤ n ( i ) ,let H i,j ∈ H i,j , M i,j := N N i ( H i,j ) , ∆ i,j := { x ∈ V \ { } | ( G i ) x ≤ H i,j } , and suppose that the M i,j -orbits on ∆ i,j are x M i,j i,j,k , ≤ k ≤ s ( i, j ) , where x i,j,k ∈ ∆ i,j for each k . For ≤ i ≤ ℓ , ≤ j ≤ n ( i ) and ≤ k ≤ s ( i, j ) , set D i,j,k := ( V, L T ⋊ G i i,j,k ) and Ω i,j,k := ( , L i,j,k ) T ⋊ G i , where L i,j,k := P i,j,k ∪ { } with P i,j,k := x H i,j i,j,k . Then { ( T ⋊ G i , D i,j,k , Ω i,j,k ) | ≤ i ≤ ℓ, ≤ j ≤ n ( i ) , ≤ k ≤ s ( i, j ) } is a representative subset of Π( d, p ) . Magma , we have computed a representative subset S of each of the following sets:Π , := { ( G, D , Ω) ∈ Π(6 , | G ∼ = SL(2 , } , Π , := { ( G, D , Ω) ∈ Π(4 , | G ☎ SL(2 , } , Π , := { ( G, D , Ω) ∈ Π(4 , | G ☎ E } , Π ,p := { ( G, D , Ω) ∈ Π(2 , p ) | G ☎ SL(2 ,
5) or G ☎ SL(2 , } , where E is an extraspecial group of order 32 and p = 5 , , , , , ,
59. Note that D is a2-( u, r + 1 , λ ) design in each case, where u = 3 for Π , , u = 3 for Π , and Π , , and u = p forΠ ,p . Our computational results are summarized in Tables 5-14, where the first row in each tablegives the common value of | G | for those proper triples in S whose group entries are pairwiseconjugate in AGL( d, p ), and the last row gives the number of proper triples in S whose r and λ values are the same and the group entries are pairwise conjugate in AGL( d, p ).In the case where G ∼ = SL(2 ,
13) and u = 3 , our computational results are given in Table 5,and we find that the corresponding G -flag graphs are all disconnected. In addition, by [22, p.73]we know that λ = 1 if and only if one of the following occurs: r = 2 and D = AG(6 , r = 8and D is one of the two Hering designs [19]; r = 26 and D is the Hering affine plane of order 27(see [18] and [10, p.236]). Table 5. Representatives of Π , | G | r + 1 3 5 9 9 14 27 27 53 λ G ☎ SL(2 , d = 4 and p = 3, the results are given in Table 6 and thecorresponding G -flag graphs are all disconnected. In addition, by [22], λ = 1 if and only if oneof the following occurs: r = 8 and D = AG(2 , r = 8 and D is the exceptional nearfield plane(see [10, pp.214, 232, 236]); r = 2 and D = AG(4 , , | G |
240 480 480 960 r + 1 3 5 9 17 41 3 5 9 41 3 5 9 41 3 5 9 41 λ G ☎ E , d = 4 and p = 3, where E is an extraspecial group of order 32, theresults are give in Table 7 and the corresponding G -flag graphs are all disconnected. By [22], λ = 1 if and only if one of the following occurs: r = 8 and D is the exceptional nearfield plane(see [10, pp.214, 232, 236]); r = 2 and D = AG(4 , , | G |
160 320 640 1920 3840 r + 1 3 5 9 9 17 3 5 9 9 17 3 5 9 17 3 9 17 3 9 17 λ G ☎ SL(2 ,
5) or G ☎ SL(2 , d = 2 and p = 5 , , , , ,
29 or279, the results are given in Tables 8-14, respectively, and the corresponding G -flag graphs areall disconnected. By [22], λ = 1 if and only if D = AG(2 , p ).Table 8. Representatives of Π , | G |
24 48 96 120 240 480 r + 1 3 4 5 5 7 9 3 5 5 9 3 5 9 3 5 3 5 3 5 λ , | G |
48 144 r + 1 3 4 5 7 7 9 13 17 25 3 4 7 9 13 25 λ , | G |
120 120 r + 1 3 4 5 6 7 9 11 11 13 21 25 3 4 5 6 7 9 11 16 21 25 31 41 λ | G |
240 600 r + 1 3 4 5 6 7 9 11 16 21 25 31 41 3 6 11 21 λ Table 11. Representatives of Π , | G | r + 1 3 4 7 9 10 13 19 25 37 73 121 λ , | G | r + 1 3 4 5 7 9 12 13 17 23 25 34 45 49 67 89 133 177 265 λ Acknowledgements
T. Fang was supported by the China Scholarship Council, X. G. Fangby National Science Foundation of China (NSFC 11231008), and S. Zhou by the AustralianResearch Council (FT110100629) and the MRGSS of the University of Melbourne. The authorswould like to thank the anonymous referees for their comments that lead to improvements ofpresentation. 28able 13. Representatives of Π , | G | r + 1 3 4 5 6 7 8 9 11 13 15 21 22 25 29 29 36 43 57 71 85 121 141 169 λ | G | r + 1 3 4 5 6 7 8 9 11 13 15 21 22 25 29 29 36 43 57 71 85 121 141 169 λ Table 14. Representatives of Π , | G | r + 1 3 4 5 6 7 9 11 13 21 25 30 59 88 117 121 146 175 233 291 349 581 697 λ References [1] E. Artin, The orders of the linear groups,
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The following
Magma codes were used in the case where G ☎ SL(2 ,
5) or G ☎ SL(2 , d = 2and p = 5 , , , , ,
29 or 59 in Section 4.8. For other cases in Section 4.8, the
Magma codesare similar, and we just give the filtering conditions by which we get the groups in Section 4.8(see “ / ∗ . . . ∗ / ” parts in the following Magma codes). / ∗ Searching for a representative subset of Π ,p ∗ / d:=2;for p in [5,7,11,19,23,29,59] dou:=p^d; A,V:=AGL(d,p); C:=GSet(A); g:=A.1;T:=NormalSubgroups(A:OrderEqual:=u); T:=T[1]‘subgroup;X:=Stabilizer(A,1);/* The following three lines of codes are called the filteringconditions */L:=Subgroups(X:OrderMultipleOf:=p^2-1);L:=[a‘subgroup:a in L| f [1,2]^y eq [2,1] then g:=y;"Find an element in G interchanging",V[1]," and ",V[2];"";break y;end if;end for;z:=g;j:=1; s:= The filtering conditions in searching for a representative subset of Π , are as follows:32 :=Subgroups(X:OrderEqual:=2184);L:=[a‘subgroup:a in L| The filtering conditions in searching for a representative subset of Π , are as follows: L:=Subgroups(X:OrderMultipleOf:=80);L:=[a‘subgroup:a in L|