The Non-Existence of Block-Transitive Subspace Designs
aa r X i v : . [ m a t h . C O ] F e b The Non-Existence of Block-Transitive SubspaceDesigns ∗Daniel R. Hawtin Jesse Lansdown Department of Mathematics, University of RijekaRijeka, Croatia, 51000. [email protected] Centre for the Mathematics of Symmetry and Computation,The University of Western AustraliaPerth, Western Australia, 6009. [email protected] 11, 2021
Abstract
Let q be a prime power and V ∼ = F nq . A t - ( n, k, λ ) q design , or simply a subspace design ,is a pair D = ( V, B ) , where B is a subset of the set of all k -dimensional subspaces of V , withthe property that each t -dimensional subspace of V is contained in precisely λ elements of B . Subspace designs are the q -analogues of balanced incomplete block designs. Such adesign is called block-transitive if its automorphism group Aut( D ) acts transitively on B . Itis shown here that if t > and D = ( V, B ) is a block-transitive t - ( n, k, λ ) q design then D is trivial, that is, B is the set of all k -dimensional subspaces of V . Tits [28] suggested that combinatorics of sets could be regarded as the limiting case q → ofcombinatorics of vector spaces over the finite field F q . Taking a combinatorial property expressedin terms of sets and rephrasing its definition in terms of F q -vector spaces gives rise to what hasbecome known as the q -analogue of the original property. A t - ( n, k, λ ) design is a pair ( P , B ) where P is a set of size n and the elements of B are k -subsets of P , called blocks , satisfyingthe condition that every t -subset of P is contained in precisely λ blocks. The q -analogue of a t - ( n, k, λ ) design is a t - ( n, k, λ ) q design. A precise definition of t - ( n, k, λ ) q designs is given inDefinition 1.2. For brevity we often refer to t - ( n, k, λ ) designs and t - ( n, k, λ ) q designs simplyas block designs and subspace designs , respectively. Subspaces designs were first referencedin the literature by Cameron [9]. See the recent survey of Braun et al. [5] for more in-depthbackground on subspace designs. A block or subspace design is block-transitive if it admits agroup of automorphisms that act transitively on its set of blocks (see Definition 1.3 for a precisedefinition of the automorphism group of a subspace design). ∗ The first author is supported by the Croatian Science Foundation under the project 6732. The second authoracknowledges the support of the Australian Research Council Discovery Grant DP200101951. This work was sup-ported by resources provided by the Pawsey Supercomputing Centre with funding from the Australian Governmentand the Government of Western Australia. q -analogue of the Fanoplane exists then its automorphism group has size at most [20]. Indeed, our main result, The-orem 1.1, shows that non-trivial block-transitive subspace designs do not exist. Theorem 1.1.
There exist no non-trivial block-transitive t - ( n, k, λ ) q designs for t < k < n and q a prime power. The proof of Theorem 1.1 relies on work of Bamberg and Penttila [3, Theorem 3.1], whodetermined all linear groups having orders with certain divisors. Their result in turn relies onGuralnick et al. [13], who make use the Aschbacher classification of maximal subgroups of clas-sical groups [1], and the classification of finite simple groups. We consider in Section 3 eachof the cases determined by [3, Theorem 3.1] in order to obtain Theorem 1.1. The majority ofcases involve a simple analysis, with the exception of those treated separately in Section 2 . Inparticular, the case treated in Lemma 2.5 involves an exhaustive computer search.In Section 1.1 we introduce notation and preliminary results for subspace designs. In Sec-tion 1.2 we discuss the concept of a primitive divisor and set up the application of [3]. Section 2deals with several specific cases. Finally, in Section 3 we prove Theorem 1.1.
In analogy with the binomial coefficient (cid:0) nk (cid:1) we define the q -binomial coefficient , (cid:18) nk (cid:19) q = ( q n − · · · ( q n − k +1 − q k − · · · ( q − . The q -binomial coefficient is also sometimes referred to as the Gaussian coefficient . Similarly,this time in analogy with the set (cid:0) Nk (cid:1) of all k -subsets of a set N , we denote the set of all k -dimensional subspaces of a vector space V over F q by (cid:0) Vk (cid:1) q . Definition 1.2.
Given integers n, k, t , and λ , with t < k n − , a t - ( n, k, λ ) q design (orbriefly a q -design or subspace design ), is a pair D = ( V, B ) , where V = F nq and B ⊆ (cid:0) Vk (cid:1) q ,such that each element of (cid:0) Vt (cid:1) q is a subspace of precisely λ elements of B .A q -Steiner system is a t - ( n, k, λ ) q design with λ = 1 . A subspace design with t = 1 is knownas a k -covering or if additionally λ = 1 a k -spread . Since k -coverings and k -spreads have beenstudied in their own right we always assume here that t > . Note that for a (classical) t - ( n, k, λ ) design the case t > also attracts the most interest. The t - ( n, k, λ ) q design with B = (cid:0) Vk (cid:1) q isthe trivial design. The number of blocks in a t - ( n, k, λ ) q design D = ( V, B ) is given by |B| = λ (cid:0) nt (cid:1) q (cid:0) kt (cid:1) q = λ ( q n − · · · ( q n − t +1 − q k − · · · ( q k − t +1 − . f defined on V we obtain the dual design D ⊥ = ( V, B ⊥ ) of a subspace design D = ( V, B ) , where B ⊥ = { U ⊥ | U ∈ B} and for each U V we have U ⊥ = { v ∈ V | f ( u, v ) = 0 , ∀ u ∈ U } . For more details on duality, see [5, Section 2.1]. Adesign such that D = D ⊥ is called self-dual .Let V ∼ = F nq . Then the Grassmann graph J q ( n, k ) is the graph having vertex set (cid:0) Vk (cid:1) q , wheretwo vertices are adjacent precisely when they intersect in a ( k − -dimensional subspace (see[7, Section 9.3]). The automorphism group of J q ( n, k ) satisfies• Aut( J q ( n, k )) ∼ = PΓL n ( q ) when < k < n and k = n .• Aut( J q ( n, k )) ∼ = PΓL n ( q ) × C when k = n .If D = ( V, B ) is a t - ( n, k, λ ) q design then B is a subset of the vertex set of J q ( n, k ) , whichallows us to state Definition 1.3 in its present form. Note the definition below agrees with thediscussion in [5, Section 2.1], i.e., that the automorphism group of D is the stabiliser of B insidethe automorphism group PΓL n ( q ) of the lattice of subspaces of V , with the exception that theduality be counted as an automorphism when D is self-dual. Note that Definition 1.3 thus impliesthat when D is not self-dual, in particular if n = 2 k , then we may assume the automorphism groupof D is a subgroup of PΓL n ( q ) , whilst if n = 2 k we must consider the larger group PΓL n ( q ) × C ,where the group C is generated by a duality. Definition 1.3.
Let D = ( V, B ) be a t - ( n, k, λ ) q design. The automorphism group Aut( D ) of D is defined to be the setwise stabiliser of B inside the automorphism group of the Grassmanngraph J q ( n, k ) . Moreover, D is called block-transitive if Aut( D ) acts transitively on B .The following result shows that the dual of a block-transitive design is again block-transitiveand allows us to restrict our attention to the case where k n/ in Section 3. Lemma 1.4. If D is a block-transitive t - ( n, k, λ ) q design then its dual D ⊥ is a block-transitive t - ( n, n − k, λ ′ ) q design, where λ ′ = λ (cid:0) n − tk (cid:1) q / (cid:0) n − tk − t (cid:1) q .Proof. First, note that if D is self-dual then the result follows immediately. Hence we may assume G = Aut( D ) is a subgroup of PΓL n ( q ) . By [26, Lemma 4.2], we have that D ⊥ is a t - ( n, n − k, λ ′ ) q design. Let D = ( V, B ) and let H be the setwise stabiliser of B in ΓL n ( q ) . Let f be anon-singular H -invariant bilinear form on V . Note that taking f to be the standard inner productsuffices, and hence such an f always exists. The dual map U U ⊥ (see, for instance, [5,Section 2.1]) then induces a bijection between (cid:0) Vk (cid:1) q and (cid:0) Vn − k (cid:1) q and a duality automorphism τ which together define a permutational isomorphism between the action of G on B and theconjugate group G τ acting on B ⊥ . It follows that D is block-transitive if and only if D ⊥ is block-transitive.The next result is simply a special case of [26, Lemma 2.1] and allows us to assume that t = 2 in Section 3. Lemma 1.5. If D = ( V, B ) is a t - ( n, k, λ ) q design, for some t > , then D is also a - ( n, k, λ ) q design, where λ = λ (cid:0) n − t − (cid:1) q (cid:0) k − t − (cid:1) q is an integer, and |B| = λ (cid:0) nt (cid:1) q (cid:0) kt (cid:1) q = λ (cid:0) n (cid:1) q (cid:0) k (cid:1) q = λ ( q n − q n − − q k − q k − − . .2 Primitive divisors A divisor r of q e − that is coprime to each q i − for i < e is said to be a primitive divisor . The primitive part of q e − is the largest primitive divisor, and we denote the primitive part of q e − by Φ ∗ e ( q ) . Note that, since Φ ∗ ( q ) = q − is even when q is odd and q e − itself is odd if q iseven, we have that Φ ∗ e ( q ) is always odd. We then have the following. Lemma 1.6.
Let D = ( V, B ) be a block-transitive - ( n, k, λ ) q design with k n/ and let G = Aut( D ) ∩ PΓL n ( q ) . Then Φ ∗ n ( q ) · Φ ∗ n − ( q ) divides the order of G .Proof. First, note that either G is actually equal to Aut( D ) , or D is self-dual and G has index inside Aut( D ) . Since Aut( D ) acts transitively on B , it follows that |B| divides | Aut( D ) | , andhence also divides | G | . By definition we have that gcd (cid:0) Φ ∗ n ( q ) , Φ ∗ n − ( q ) (cid:1) = 1 . Moreover, Φ ∗ i ( q ) is always odd. Thus, it suffices for us to prove that Φ ∗ i ( q ) divides | G | for each i = n, n − . Now |B| = λ ( q n − q n − − q k − q k − − . Since < k and k n/ , we have that k < n − . Thus, for i = n, n − , we have that Φ ∗ i ( q ) is coprime to each of q k − and q k − − . Hence Φ ∗ n ( q )Φ ∗ n − ( q ) divides |B| , and therefore | G | ,as required.The significance of Lemma 1.6 is that it allows [3, Theorem 3.1] to be applied in Section 3. In the next section we deal with some special cases that either do not satisfy the conditionsrequired in Section 3 in order to apply [3, Theorem 3.1], or otherwise require particular attention.
Lemma 2.1.
Suppose D = ( V, B ) is a block-transitive - (6 , , λ ) design, for some λ > , andlet G ∼ = Aut( D ) ∩ PΓL (2) . Then both the size of B and the order of G are divisible by .Proof. Note that if D is self-dual, then G has index in Aut( D ) , and otherwise G = Aut( D ) .Since D is block-transitive, it follows that |B| divides | Aut( D ) | . Hence, |B| divides | G | . Now, |B| = 63 · · λ = 3 · · λ. Thus ·
31 = 93 divides the order of G . Lemma 2.2.
Suppose D = ( V, B ) is a block-transitive - (6 , , λ ) design, for some λ > , andlet G ∼ = Aut( D ) ∩ PΓL (2) . Then G does not stabilise a -dimensional subspace of V .Proof. Firstly, observe that either G = Aut( D ) , or D is self-dual and G has index in Aut( D ) .Therefore, since Aut( D ) acts transitively on B , it follows that B is either a single G -orbit or theunion of two equal-sized G -orbits, where some element of Aut( D ) \ G fuses the two G -orbits toform B . Now, by Lemma 2.1, |B| must be divisible by . In the first case, the length of the single G -orbit must thus be divisible by . In the latter case, the length of each G -orbit must be |B| / ,and since gcd(93 ,
2) = 1 , it follows that the size of both G -orbits must also be divisible by .4ssume for a contradiction that G stabilises a subspace U = h v i , for some non-zero v ∈ V . Let W = h v , . . . , v i , where v , . . . , v form a basis for V . Furthermore, let H = N ⋊ K ∼ =AGL (2) be the stabiliser of U inside SL (2) ∼ = PSL (2) ∼ = PΓL (2) , where each element of N is given, for some u ∈ W , by the map v v + u and v i v i for i = 1 , and K is given bythe natural action of SL (2) on W extended to all of V . Note that K then acts on N , and thataction is transitive on the non-identity elements of N .Consider the projection P of G into H/N ∼ = SL (2) via the homomorphism nσ σ for n ∈ N and σ ∈ K . Since is coprime to the order of N , it follows that divides the order of P . By [6, Table 8.24], there are no maximal subgroups of SL (2) having order divisible by , andhence also no proper subgroups of SL (2) having order divisible by . Therefore, P ∼ = SL (2) .Since the Schur multiplier of SL (2) is trivial and there is only one conjugacy class of groupsisomorphic to SL (2) inside H , we may assume that G contains K . If G ∩ N is the trivial groupthen G = K . If G ∩ N is non-trivial then, since the action of K is transitive on the non-identityelements of N , it follows that G ∩ N = N and G = H . We now consider the G -orbits on (cid:0) V (cid:1) for each possibility of G .Suppose G = K . Then (cid:0) W (cid:1) is one G -orbit. Note that if we consider V to be the underlyingvector space of PG (2) then W is the underlying vector space of a subgeometry PG (2) andthe points not in this subgeometry can be considered to form an affine space AG (2) . A -dimensional subspace of V that is not in (cid:0) W (cid:1) decomposes into a -dimensional subspace X of W and a -flat of AG (2) lying in the parallel class corresponding to X . Since K is alsotransitive on (cid:0) W (cid:1) , it follows that K has / = 8 orbits on the -flats of AG (2) , correspondingto a further orbits on (cid:0) V (cid:1) . Each orbit has length has length , which is not divisible by ,giving a contradiction.Now suppose G = H . Then there are just two orbits of G on (cid:0) V (cid:1) , the first being (cid:0) W (cid:1) , withlength , and the second being the set of all -dimensional subspaces that intersect W in a -dimensional subspace, with length . Neither length is divisible by , a contradiction. Lemma 2.3.
No non-trivial block-transitive - ( n, k, λ ) q design exists with q n = 2 .Proof. Suppose, for a contradiction, that such a subspace design D = ( V, B ) exists. By defini-tion we have that k > and by Lemma 1.4 we may assume that k n/ . Since q n = 2 , sothat n , it follows that k = 3 , q = 2 and n = 6 . Let G = Aut( D ) ∩ PSL (2) . By Lemma 2.1,we then have that · divides the order of G .Now, D non-trivial implies that B is not the set of all -dimensional subspaces of V , fromwhich it follows that G must be a proper subgroup of PΓL (2) ∼ = PSL (2) and is thus containedin some maximal subgroup of PSL (2) . By [6, Tables 8.24 and 8.25], there are two conjugacyclasses of such maximal subgroups that have order divisible by , a representative of eachclass being isomorphic to : GL (2) as the stabiliser of a -dimensional subspace or a -dimensional subspace of V , respectively. By Lemma 2.2 there is no block-transitive design suchthat G stabilises a -dimensional subspace. Moreover, if G were to stabilise a -dimensionalsubspace, then Aut( D ⊥ ) ∩ PSL (2) would stabilise a -dimensional subspace. However, byLemma 1.4, D ⊥ has the same parameters as D , and hence there is no such design where G stabilises a -dimensional subspace. Lemma 2.4.
No non-trivial block-transitive - ( n, k, λ ) q design exists with q n − = 2 .Proof. Suppose that such a subspace design D = ( V, B ) exists and let G = Aut( D ) . Bydefinition we have that k > and by Lemma 1.4 we may assume that k n/ . Since n we have that k = 3 , q = 2 and n = 7 . Note that n odd implies D is not self-dual and G ΓL (2) ∼ = PSL (2) . If G ∼ = PSL (2) then G acts transitively on (cid:0) V (cid:1) , which implies that B = (cid:0) V (cid:1) . However this is not the case, as D is non-trivial, and hence we deduce that G is aproper subgroup of PSL (2) . It follows from this that G is contained in some maximal subgroupof PSL (2) . Now, Lemma 1.5 implies that |B| = 3 · · λ . Since D is block-transitive it followsthat |B| must divide | G | . By [6, Tables 8.35 and 8.36], the only maximal subgroup of PSL (2) that has order divisible by is the normaliser of a Singer cycle. However, this group does nothave order divisible by , giving a contradiction. Lemma 2.5.
There does not exist a - (11 , , design having automorphism group isomorphicto ΓL (2 ) .Proof. Note that the dimension of V is odd here, which implies that D is not self-dual, andhence Aut( D ) PΓL (2) ∼ = SL (2) . By [6, Table 8.70], there is a unique conjugacy class ofsubgroups isomorphic to ΓL (2 ) in SL (2) . Thus, without loss of generality, we may construct G ∼ = PΓL (2 ) as the normaliser of a randomly found element of order in PΓL (2) . Ifa - (11 , , design were to exist with automorphism group isomorphic to G , then it must be asingle orbit of G on -spaces of F . By computer, it was found that none of the orbits of G on -spaces of F yield a - (11 , , design; see Remark 2.6 for more information. Remark 2.6.
The computation required to prove Lemma 2.5 was performed in GAP [12] withthe package FinInG [2]. Note that a - (11 , , design can equivalently be described as a setof projective -spaces of PG (2) such that every projective -space is contained in precisely elements. This formulation lends itself more naturally to construction in FinInG. Since there aretoo many (specifically, ) projective -spaces to reasonably fit in memory we insteadconstructed orbits of G by finding suitably many distinct and unique representatives from eachorbit. Representatives were determined uniquely by choosing them to be lexicographically leastin their orbits. These representatives, as well as GAP code, are made available for ease ofverification at [15]. In this section we prove Theorem 1.1 by applying [3, Theorem 3.1]. We frequently reference thecases, which are related to the Aschbacher classes as in [1], in the manner that they are listedin [3]. These cases are organised according to the following categories: classical , reducible , imprimitive , extension field , symplectic , and nearly simple . For information regarding specificgroups see, for instance, [29]. Remark 3.1.
Before continuing, we note that there is a very small error in the statement of theextension field case in [3, Theorem 3.1]. Case (a) of the extension field case does not requirethat b be a non-trivial divisor of gcd( d, e ) ; see [13, Example 2.4] for clarification regarding theconditions in this case.The next result splits the treatment of - ( n, k, λ ) q designs into three cases. Lemma 3.2.
Let q = p f , where p is prime, let D = ( V, B ) be a - ( n, k, λ ) q design, and let G bethe setwise stabiliser of B inside ΓL n ( q ) . Then one of the following holds:1. One of Φ ∗ nf ( p ) or Φ ∗ ( n − f ( p ) is trivial.2. Both of Φ ∗ nf ( p ) and Φ ∗ ( n − f ( p ) are non-trivial and divide the order of G where G ΓL ( q n ) as in the extension field case a) of [3, Theorem 3.1]. . Both of Φ ∗ nf ( p ) and Φ ∗ ( n − f ( p ) are non-trivial and divide the order of ˆ G = G ∩ GL n ( q ) , and ˆ G is as in one of the classical, reducible, imprimitive, symplectic, or nearly simple casesof [3, Theorem 3.1].Proof. If either Φ ∗ nf ( p ) = 1 or Φ ∗ ( n − f ( p ) = 1 then part 1 holds. Suppose both of Φ ∗ nf ( p ) and Φ ∗ ( n − f ( p ) are non-trivial. Let H = G/ ( G ∩ Z (GL n ( q ))) . Note that if D is self-dual then it maybe the case that H has equal-sized orbits on B . Since H is a quotient of G , it follows fromLemma 1.6 that Φ ∗ n ( q ) · Φ ∗ n − ( q ) divides the order of G . Now, for any integer e > we havethat Φ ∗ ef ( p ) divides Φ ∗ e ( q ) , and hence both Φ ∗ nf ( p ) and Φ ∗ ( n − f ( p ) divide the order of G . Hence[3, Theorem 3.1] applies. Note that F q and F fp are isomorphic as F p -vector spaces, and so V may also be considered to be a vector space over F p of dimension nf . Thus, when applying [3,Theorem 3.1] we consider G to be a subgroup of GL nf ( p ) .If G ΓL ( q n ) as in the extension field case a) of [3, Theorem 3.1], then part 2 holds.Suppose that G falls under extension field case b), so that G ΓL nf/s ( p s ) for some integer s , with < s < nf and s dividing both nf and ( n − f . This implies that s divides f . Byassumption, we have that G ΓL n ( q ) , and hence s = f . If f = 1 , this gives a contradiction,and hence G is a classical, reducible, imprimitive, symplectic, or nearly simple example, as inpart 3. If f > then the conditions of the extension field case imply that Φ ∗ nf ( p ) and Φ ∗ ( n − f ( p ) divide the order of ˆ G = G ∩ GL n ( q ) , and ˆ G is as in one of the classical, reducible, imprimitive,symplectic, or nearly simple cases of [3, Theorem 3.1], treated now as a subgroup of GL n ( q ) ,and part 3 holds.Next we consider the case that part 2 of Lemma 3.2 holds. Lemma 3.3.
Let q = p f , let D = ( V, B ) be a - ( n, k, λ ) q design, and let G be the stabiliserof B inside ΓL n ( q ) . Moreover, suppose that G ΓL ( q n ) and both of Φ ∗ nf ( p ) , Φ ∗ ( n − f ( p ) arenon-trivial and divide the order of G . Then D is not block-transitive.Proof. Suppose that D is block-transitive. Then, since Φ ∗ nf ( p ) and Φ ∗ ( n − f ( p ) are non-trivial anddivide the order of G , and G is a subgroup of GL nf ( p ) , we may apply [3, Theorem 3.1], in whichcase G falls under the extension field case a). Since n > , we have that p d = 2 , , , or . Note that, since d is odd in each of these cases, we have that D is not self-dual. Hence Aut( D ) is the quotient of G by G ∩ Z (GL n ( q )) , where Z (GL n ( q )) is the centre of GL n ( q ) .Thus | Aut( D ) | divides d ( p d − / ( p − . Since Aut( D ) acts transitively on B , the size of B must divide the order of Aut( D ) . Now, |B| = λ ( p d − p d − − p k − p k − − , for some integers λ and k with λ > and k d/ . Hence the following is an integer: λ | Aut( D ) ||B| = d ( p d − p −
1) ( p k − p k − − p d − p d − −
1) = d ( p k − p k − − p d − − p − . This is true only if p d = 2 with k = 5 and λ = 1 or ; or p d = 3 with k = 3 and λ = 1 .By [5, Theorem 2], the derived design of a - (11 , , design would be a - (10 , , design.However, by [5, Lemma 4], no - (10 , , design exists, in particular, such a design wouldbe a spread and k = 4 does not divide n = 10 . For the case of a - (7 , , design, thedivisibility condition above implies that Aut( D ) = PΓL ( p d ) , in which case we may assume that G = ΓL (3 ) . However, by [21, Theorem 2 (3)], no - (7 , , design with G acting transitively7n V \ { } exists. Thus q = 2 , n = 11 and k = λ = 5 . However, Lemma 2.5 rules out theexistence of such a design.The classical cases are excluded by the following, Lemma 3.4. Lemma 3.4.
Let q = p f , let D = ( V, B ) be a block-transitive - ( n, k, λ ) q design, let G be thestabiliser of B inside ΓL n ( q ) , and let ˆ G = G ∩ GL n ( q ) . Moreover, suppose that both of Φ ∗ nf ( p ) and Φ ∗ ( n − f ( p ) are non-trivial and divide the order of ˆ G . Then none of the following hold:(a) SL n ( q ) E ˆ G ,(b) Sp n ( q ) E ˆ G (c) q is a square and SU n ( q / ) E ˆ G ,(d) Ω ǫn ( q ) E ˆ G where ǫ = ± for n even, and ǫ = ◦ when nq odd.Proof. For (a), SL n ( q ) acts transitively on the set of all k -spaces of V , for any choice of k , sothat in this case the only subspace design invariant under ˆ G is the trivial design.For case (b) we have that ˆ G contains Sp n ( q ) as a normal subgroup but does not containany field automorphisms. Since n > , the only automorphisms of Sp n ( q ) we need to considerare those induced by the centre of GL n ( q ) , and hence the order of ˆ G is at most twice the orderof Sp n ( q ) (see [29, Section 3.5.5]). Hence | ˆ G | divides q n / Q n/ i =1 ( q i − ; we claim that thisimplies that | ˆ G | is not divisible by Φ ∗ ( n − f ( p ) . Recall that Φ ∗ e ( p ) is odd for all e > so that thefactor of is inconsequential. Moreover, q n / is coprime to Φ ∗ ( n − f ( p ) . Thus, we need only beinterested in the factors ( q i − of | ˆ G | and these only appear for even i . Since n − is odd, theclaim holds and this case does not occur.Consider now case (c). By [3, Theorem 3.1], we have that Φ ∗ e ( p ) divides the order of thenormaliser of SU n ( q / ) only when e is odd, but e must be able to take both the values nf and ( n − f here, at least one of which is even.For case (d), we have that if n is even then ˆ G GO ǫn ( q ) where ǫ = + or − , and if n is oddthen ˆ G GO n ( q ) . Referring to Table 1, we see that Φ ∗ ( n − f ( p ) does not divide the order of GO ǫn ( q ) (it should be noted here though that Φ ∗ nf ( p ) divides q n + 1 , and hence does divide | ˆ G | ).Also, Φ ∗ nf ( p ) does not divide the order of GO n ( q ) . Hence this case does not occur.group order GO m +1 ( q ) 2 q m ( q − q − · · · ( q m − +2 m ( q ) 2 q m ( m − ( q − q − · · · ( q m − − q m − − m ( q ) 2 q m ( m − ( q − q − · · · ( q m − − q m + 1) Table 1: Orders of orthogonal groups.The reducible cases are excluded by the following, Lemma 3.5.
Lemma 3.5.
Let q = p f , let D = ( V, B ) be a block-transitive - ( n, k, λ ) q design, and let G bethe stabiliser of the B inside ΓL n ( q ) , and let ˆ G = G ∩ GL n ( q ) . Then ˆ G H ∼ = q m ( n − m ) · (GL m ( q ) × GL n − m ( q )) for some m such that < m < n . roof. By definition, Φ ∗ nf ( p ) is coprime to each factor q i − dividing | H | , that is, for i max { m, n − m } < n . Since Φ ∗ nf ( p ) divides q n − , it follows that Φ ∗ nf ( p ) is also coprimeto q m ( n − m ) , and hence does not divide the order of ˆ G .We are now able to provide the proof of Theorem 1.1. Proof of Theorem 1.1.
Suppose D = ( V, B ) is a non-trivial block-transitive t - ( n, k, λ ) q design,where V = F nq , and let q = p f , where p is prime. By Lemma 1.5, we may assume that t = 2 , andhence that k > . By Lemma 1.4, we may assume that k n/ , so that n > . Let G be thesetwise stabiliser of B inside ΓL n ( q ) . We may then apply Lemma 3.2. For part 1 of Lemma 3.2,Zsigmondy’s theorem [30] states that if Φ ∗ e ( p ) = 1 then p e = 2 . However, Lemmas 2.3 and 2.4rule out the existence of any - ( n, k, λ ) q design with q n or q n − equal to . Lemma 3.3 rulesout the occurrence of part 2 of Lemma 3.2. This leaves the possibility that part 3 of Lemma 3.2holds, that is, Φ ∗ nf ( p ) and Φ ∗ ( n − f ( p ) divide the order of ˆ G = G ∩ GL n ( q ) , and ˆ G is as in oneof the classical, reducible, imprimitive, symplectic, or nearly simple cases of [3, Theorem 3.1].The classical examples are ruled out by Lemma 3.4, and the reducible examples are ruledout by Lemma 3.5. The remaining imprimitive, symplectic type, and nearly simple cases areimmediately excluded by divisibility conditions, since non-trivial Φ ∗ nf ( p ) and Φ ∗ ( n − f ( p ) are re-quired to divide the order of G ∩ GL n ( q ) ; see Remark 3.6 for more detail. This completes theproof of Theorem 1.1. Remark 3.6.
For the benefit of the reader looking to verify our results in the imprimitive, sym-plectic type, and nearly simple examples, we recall now that n > ( d in the tables of [3]) andthat the variable e in [3] must be able to take both the values nf and ( n − f for the same groupaction. In particular, many of the groups in the tables of [3] only allow e to take one specific valuein a given group action, allowing such a group to be immediately ruled out. The reader shouldalso note that the meaning of the variable n as used in [3] for the alternating group case is not thesame as the meaning of n here. Having noted these requirements, these cases are completedby observing that there is no group for these cases such that e can assume each of the values nf and ( n − f . References [1] M. Aschbacher. On the maximal subgroups of the finite classical groups.
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