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Theorem 1.1. There exist no non-trivial block-transitive t-(n,k,λ)q designs for 2 6 t The proof of Theorem 1.1 relies on work of Bamberg and Penttila [3, Theorem 3.1], who determined all linear groups having orders with certain divisors. Their result in turn relies on Guralnick 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 each of the cases determined by [3, Theorem 3.1] in order to obtain Theorem 1.1. The majority of cases involve a simple analysis, with the exception of those treated separately in Section 2 . In particular, 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 2 deals with several specific cases. Finally, in Section 3 we prove Theorem 1.1. 1.1 Subspace designs n In analogy with the binomial coefficient k we define the q-binomial coefficient, − n (qn − 1) ··· (qn k+1 − 1) = k . kq (q − 1) ··· (q − 1) The q-binomial coefficient is also sometimes referred to as the Gaussian coefficient. Similarly, N this time in analogy with the set k of all k-subsets of a set N, we denote the set of all k- dimensional subspaces of a vector space over F by V . V q k q Definition 1.2. Given integers n, k, t, and λ, with 1 6 t If D = (V, B) is a t-(n,k,λ)q design then B is a subset of the vertex set of Jq(n, k), which allows us to state Definition 1.3 in its present form. Note the definition below agrees with the discussion in [5, Section 2.1], i.e., that the automorphism group of D is the stabiliser of B inside the automorphism group PΓLn(q) of the lattice of subspaces of V , with the exception that the duality be counted as an automorphism when D is self-dual. Note that Definition 1.3 thus implies that when D is not self-dual, in particular if n =26 k, then we may assume the automorphismgroup of D is a subgroup of PΓLn(q), whilst if n =2k we must consider the larger group PΓLn(q)×C2, where the group C2 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 Grassmann graph Jq(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-transitive and allows us to restrict our attention to the case where k 6 n/2 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 ) design, where λ = λ n t / n t . q k q k−t 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 PΓLn(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 ΓLn(q). Let f be a non-singular H-invariant bilinear form on V . Note that taking f to be the standard inner product suffices, and hence such an f always exists. The dual map U 7→ U ⊥ (see, for instance, [5, Section 2.1]) then induces a bijection between V and V and a duality automorphism k q n−k q τ which together define a permutational isomorphism between the action of G on B and the conjugate group Gτ 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 > 2, then D is also a 2-(n,k,λ2)q design, where n−2 t−2 q λ2 = λ k−2 t−2 q is an integer, and n n (qn − 1)(qn−1 − 1) |B| = λ t q = λ 2 q = λ . k 2 k 2 (qk − 1)(qk−1 − 1) t q 2 q 3 1.2 Primitive divisors A divisor r of qe − 1 that is coprime to each qi − 1 for i < e is said to be a primitive divisor. The primitive part of qe − 1 is the largest primitive divisor, and we denote the primitive part of qe − 1 ∗ ∗ e by Φe(q). Note that, since Φ1(q) = q − 1 is even when q is odd and q − 1 itself is odd if q is ∗ even, we have that Φe(q) is always odd. We then have the following. Lemma 1.6. Let D = (V, B) be a block-transitive 2-(n,k,λ)q design with k 6 n/2 and let ∗ ∗ G = Aut(D) ∩ PΓLn(q). Then Φn(q) · Φn−1(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 2 inside Aut(D). Since Aut(D) acts transitively on B, it follows that |B| divides | Aut(D)|, and hence also divides 2|G|. By definition we have that ∗ ∗ gcd Φn(q), Φn−1(q) =1. ∗ ∗ Moreover, Φi (q) is always odd. Thus, it suffices for us to prove that Φi (q) divides 2|G| for each i = n, n − 1. Now (qn − 1)(qn−1 − 1) |B| = λ . (qk − 1)(qk−1 − 1) ∗ Since 2 The significance of Lemma 1.6 is that it allows [3, Theorem 3.1] to be applied in Section 3. 2 Special cases In the next section we deal with some special cases that either do not satisfy the conditions required 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 2-(6, 3,λ)2 design, for some λ > 1, and ∼ let G = Aut(D) ∩ PΓL6(2). Then both the size of B and the order of G are divisible by 93. Proof. Note that if D is self-dual, then G has index 2 in Aut(D), and otherwise G = Aut(D). Since D is block-transitive, it follows that |B| divides | Aut(D)|. Hence, |B| divides 2|G|. Now, 63 · 31 |B| = λ =3 · 31 · λ. 7 · 3 Thus 3 · 31 = 93 divides the order of G. Lemma 2.2. Suppose D =(V, B) is a block-transitive 2-(6, 3,λ)2 design, for some λ > 1, and ∼ let G = Aut(D) ∩ PΓL6(2). Then G does not stabilise a 1-dimensional subspace of V . Proof. Firstly, observe that either G = Aut(D), or D is self-dual and G has index 2 in Aut(D). Therefore, since Aut(D) acts transitively on B, it follows that B is either a single G-orbit or the union of two equal-sized G-orbits, where some element of Aut(D)\G fuses the two G-orbits to form B. Now, by Lemma 2.1, |B| must be divisible by 93. In the first case, the length of the single G-orbit must thus be divisible by 93. In the latter case, the length of each G-orbit must be |B|/2, and since gcd(93, 2)=1, it follows that the size of both G-orbits must also be divisible by 93. 4 Assume for a contradiction that G stabilises a subspace U = hv1i, for some non-zero v1 ∈ ∼ V . Let W = hv2,...,v6i, where v1,...,v6 form a basis for V . Furthermore, let H = N ⋊ K = ∼ ∼ AGL5(2) be the stabiliser of U inside SL6(2) = PSL6(2) = PΓL6(2), where each element of N is given, for some u ∈ W , by the map v1 7→ v1 + u and vi 7→ vi for i =16 , and K is given by the natural action of SL5(2) on W extended to all of V . Note that K then acts on N, and that action is transitive on the non-identity elements of N. ∼ Consider the projection P of G into H/N = SL5(2) via the homomorphism nσ 7→ σ for n ∈ N and σ ∈ K. Since 93 is coprime to the order of N, it follows that 93 divides the order of P . By [6, Table 8.24], there are no maximal subgroupsof SL5(2) having order divisible by 93, and ∼ hence also no proper subgroups of SL5(2) having order divisible by 93. Therefore, P = SL5(2). Since the Schur multiplier of SL5(2) is trivial and there is only one conjugacy class of groups isomorphic to SL5(2) inside H, we may assume that G contains K. If G ∩ N is the trivial group then G = K. If G ∩ N is non-trivial then, since the action of K is transitive on the non-identity elements of , it follows that and . We now consider the -orbits on V N G ∩ N = N G = H G 3 2 for each possibility of G. Suppose . Then W is one -orbit. Note that if we consider to be the underlying G = K 3 2 G V vector space of PG5(2) then W is the underlying vector space of a subgeometry PG4(2) and the points not in this subgeometry can be considered to form an affine space AG5(2). A 3- dimensional subspace of that is not in W decomposes into a -dimensional subspace V 3 2 2 X of W and a 2-flat of AG5(2) lying in the parallel class corresponding to X. Since K is also transitive on W , it follows that has 5 2 orbits on the -flats of , corresponding 2 2 K 2 /2 =8 2 AG5(2) to a further orbits on V . Each orbit has length has length , which is not divisible by , 8 3 2 155 93 giving a contradiction. Now suppose . Then there are just two orbits of on V , the first being W , with G = H G 3 2 3 2 length 155, and the second being the set of all 3-dimensional subspaces that intersect W in a 2-dimensional subspace, with length 1240. Neither length is divisible by 93, a contradiction. n 6 Lemma 2.3. No non-trivial block-transitive 2-(n,k,λ)q design exists with q =2 . Proof. Suppose, for a contradiction, that such a subspace design D =(V, B) exists. By defini- tion we have that k > 3 and by Lemma 1.4 we may assume that k 6 n/2. Since qn = 26, so that n 6 6, it follows that k =3, q =2 and n =6. Let G = Aut(D) ∩ PSL6(2). By Lemma 2.1, we then have that 3 · 31 divides the order of G. Now, D non-trivial implies that B is not the set of all 3-dimensional subspaces of V , from ∼ which it follows that G must be a proper subgroup of PΓL6(2) = PSL6(2) and is thus contained in some maximal subgroup of PSL6(2). By [6, Tables 8.24 and 8.25], there are two conjugacy classes of such maximal subgroups that have order divisible by 31, a representative of each 5 class being isomorphic to 2 : GL5(2) as the stabiliser of a 1-dimensional subspace or a 5- dimensional subspace of V , respectively. By Lemma 2.2 there is no block-transitive design such that G stabilises a 1-dimensional subspace. Moreover, if G were to stabilise a 5-dimensional ⊥ subspace, then Aut(D ) ∩ PSL6(2) would stabilise a 1-dimensional subspace. However, by Lemma 1.4, D⊥ has the same parameters as D, and hence there is no such design where G stabilises a 5-dimensional subspace. n−1 6 Lemma 2.4. No non-trivial block-transitive 2-(n,k,λ)q design exists with q =2 . Proof. Suppose that such a subspace design D = (V, B) exists and let G = Aut(D). By definition we have that k > 3 and by Lemma 1.4 we may assume that k 6 n/2. Since n 6 7 we have that k = 3, q = 2 and n = 7. Note that n odd implies D is not self-dual and G 6 5 ∼ . If ∼ then acts transitively on V , which implies that PΓL7(2) = PSL7(2) G = PSL7(2) G 3 2 V . However this is not the case, as is non-trivial, and hence we deduce that is a B = 3 2 D G proper subgroup of PSL7(2). It follows from this that G is contained in some maximal subgroup of PSL7(2). Now, Lemma 1.5 implies that |B| =3 · 127 · λ. Since D is block-transitive it follows that |B| must divide |G|. By [6, Tables 8.35 and 8.36], the only maximal subgroup of PSL7(2) that has order divisible by 127 is the normaliser of a Singer cycle. However, this group does not have order divisible by 3, giving a contradiction. Lemma 2.5. There does not exist a 2-(11, 5, 5)2 design having automorphism group isomorphic 11 to ΓL1(2 ). Proof. Note that the dimension of V is odd here, which implies that D is not self-dual, and ∼ hence Aut(D) 6 PΓL11(2) = SL11(2). By [6, Table 8.70], there is a unique conjugacy class of 11 subgroups isomorphic to ΓL1(2 ) in SL11(2). Thus, without loss of generality, we may construct ∼ 11 G = PΓL1(2 ) as the normaliser of a randomly found element of order 2047 in PΓL11(2). If a 2-(11, 5, 5)2 design were to exist with automorphism group isomorphic to G, then it must be a F11 single orbit of G on 5-spaces of 2 . By computer, it was found that none of the 157607 orbits of F11 G on 5-spaces of 2 yield a 2-(11, 5, 5)2 design; see Remark 2.6 for more information. Remark 2.6. The computation required to prove Lemma 2.5 was performed in GAP [12] with the package FinInG [2]. Note that a 2-(11, 5, 5)2 design can equivalently be described as a set of projective 4-spaces of PG10(2) such that every projective 1-space is contained in precisely 5 elements. This formulation lends itself more naturally to construction in FinInG. Since there are too many (specifically, 3548836819) projective 4-spaces to reasonably fit in memory we instead constructed orbits of G by finding suitably many distinct and unique representatives from each orbit. Representatives were determined uniquely by choosing them to be lexicographically least in their orbits. These representatives, as well as GAP code, are made available for ease of verification at [15]. 3 Main results In this section we prove Theorem 1.1 by applying [3, Theorem 3.1]. We frequently reference the cases, which are related to the Aschbacher classes as in [1], in the manner that they are listed in [3]. These cases are organised according to the following categories: classical, reducible, imprimitive, extension field, symplectic, and nearly simple. For information regarding specific groups see, for instance, [29]. Remark 3.1. Before continuing, we note that there is a very small error in the statement of the extension field case in [3, Theorem 3.1]. Case (a) of the extension field case does not require that b be a non-trivial divisor of gcd(d, e); see [13, Example 2.4] for clarification regarding the conditions in this case. The next result splits the treatment of 2-(n,k,λ)q designs into three cases. f Lemma 3.2. Let q = p , where p is prime, let D =(V, B) be a 2-(n,k,λ)q design, and let G be the setwise stabiliser of B inside ΓLn(q). Then one of the following holds: ∗ ∗ 1. One of Φnf (p) or Φ(n−1)f (p) is trivial. ∗ ∗ 2. Both of Φnf (p) and Φ(n−1)f (p) are non-trivial and divide the order of G where G 6 n ΓL1(q ) as in the extension field case a) of [3, Theorem 3.1]. 6 ∗ ∗ ˆ 3. Bothof Φnf (p) and Φ(n−1)f (p) are non-trivial and divide the order of G = G∩GLn(q), and Gˆ is as in one of the classical, reducible, imprimitive, symplectic, or nearly simple cases of [3, Theorem 3.1]. ∗ ∗ ∗ Proof. If either Φnf (p)=1 or Φ(n−1)f (p)=1 then part 1 holds. Suppose both of Φnf (p) and ∗ Φ(n−1)f (p) are non-trivial. Let H = G/ (G ∩ Z(GLn(q))). Note that if D is self-dual then it may be the case that H has 2 equal-sized orbits on B. Since H is a quotient of G, it follows from ∗ ∗ Lemma 1.6 that Φn(q) · Φn−1(q) divides the order of G. Now, for any integer e > 1 we have ∗ ∗ ∗ ∗ that Φef (p) divides Φe(q), and hence both Φnf (p) and Φ(n−1)f (p) divide the order of G. Hence F Ff F [3, Theorem 3.1] applies. Note that q and p are isomorphic as p-vector spaces, and so V may also be considered to be a vector space over Fp of dimension nf. Thus, when applying [3, Theorem 3.1] we consider G to be a subgroup of GLnf (p). n If G 6 ΓL1(q ) as in the extension field case a) of [3, Theorem 3.1], then part 2 holds. s Suppose that G falls under extension field case b), so that G 6 ΓLnf/s(p ) for some integer s, with 1 < s < nf and s dividing both nf and (n − 1)f. This implies that s divides f. By assumption, we have that G 6 ΓLn(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 in ∗ ∗ part 3. If f > 2 then the conditions of the extension field case imply that Φnf (p) and Φ(n−1)f (p) divide the order of Gˆ = G ∩ GLn(q), and Gˆ 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 GLn(q), and part 3 holds. Next we consider the case that part 2 of Lemma 3.2 holds. f Lemma 3.3. Let q = p , let D = (V, B) be a 2-(n,k,λ)q design, and let G be the stabiliser n ∗ ∗ of B inside ΓLn(q). Moreover, suppose that G 6 ΓL1(q ) and both of Φnf (p), Φ(n−1)f (p) are non-trivial and divide the order of G. Then D is not block-transitive. ∗ ∗ Proof. Suppose that D is block-transitive. Then, since Φnf (p) and Φ(n−1)f (p) are non-trivial and divide the order of G, and G is a subgroup of GLnf (p), we may apply [3, Theorem 3.1], in which case G falls under the extension field case a). Since n > 6, we have that pd = 211, 213, 219, 37 or 57. 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(GLn(q)), where Z(GLn(q)) is the centre of GLn(q). Thus | Aut(D)| divides d(pd − 1)/(p − 1). Since Aut(D) acts transitively on B, the size of B must divide the order of Aut(D). Now, (pd − 1)(pd−1 − 1) |B| = λ , (pk − 1)(pk−1 − 1) for some integers λ and k with λ > 1 and 3 6 k 6 d/2. Hence the following is an integer: | Aut(D)| d(pd − 1) (pk − 1)(pk−1 − 1) d(pk − 1)(pk−1 − 1) λ = = . |B| (p − 1) (pd − 1)(pd−1 − 1) (pd−1 − 1)(p − 1) This is true only if pd = 211 with k = 5 and λ = 1 or 5; or pd = 37 with k = 3 and λ = 1. By [5, Theorem 2], the derived design of a 2-(11, 5, 1)2 design would be a 1-(10, 4, 1)2 design. However, by [5, Lemma 4], no 1-(10, 4, 1)2 design exists, in particular, such a design would be a spread and k = 4 does not divide n = 10. For the case of a 2-(7, 3, 1)3 design, the d divisibility condition above implies that Aut(D) = PΓL1(p ), in which case we may assume that 7 G = ΓL1(3 ). However, by [21, Theorem 2 (3)], no 2-(7, 3, 1)3 design with G acting transitively 7 on V \{0} exists. Thus q = 2, n = 11 and k = λ = 5. However, Lemma 2.5 rules out the existence of such a design. The classical cases are excluded by the following, Lemma 3.4. f Lemma 3.4. Let q = p , let D = (V, B) be a block-transitive 2-(n,k,λ)q design, let G be the ˆ ∗ stabiliser of B inside ΓLn(q), and let G = G ∩ GLn(q). Moreover, suppose that both of Φnf (p) ∗ ˆ and Φ(n−1)f (p) are non-trivial and divide the order of G. Then none of the following hold: (a) SLn(q) E Gˆ, ˆ (b) Spn(q) E G 1/2 (c) q is a square and SUn(q ) E Gˆ, ǫ ˆ (d) Ωn(q) E G where ǫ = ± for n even, and ǫ = ◦ when nq odd. Proof. For (a), SLn(q) acts transitively on the set of all k-spaces of V , for any choice of k, so that in this case the only subspace design invariant under Gˆ is the trivial design. ˆ For case (b) we have that G contains Spn(q) as a normal subgroup but does not contain any field automorphisms. Since n > 6, the only automorphisms of Spn(q) we need to consider are those induced by the centre of GLn(q), and hence the order of Gˆ is at most twice the order ˆ n2/4 n/2 2i of Spn(q) (see [29, Section 3.5.5]). Hence |G| divides 2q i=1(q − 1); we claim that this ˆ ∗ ∗ implies that |G| is not divisible by Φ(n−1)f (p). Recall that Φe(pQ) is odd for all e > 1 so that the n2/4 ∗ factor of 2 is inconsequential. Moreover, q is coprime to Φ(n−1)f (p). Thus, we need only be interested in the factors (qi − 1) of |Gˆ| and these only appear for even i. Since n − 1 is odd, the claim holds and this case does not occur. ∗ Consider now case (c). By [3, Theorem 3.1], we have that Φe(p) divides the order of the 1/2 normaliser of SUn(q ) only when e is odd, but e must be able to take both the values nf and (n − 1)f here, at least one of which is even. ˆ ǫ For case (d), we have that if n is even then G 6 GOn(q) where ǫ =+ or −, and if n is odd ˆ ∗ then G 6 GOn(q). Referring to Table 1, we see that Φ(n−1)f (p) does not divide the order of ǫ ∗ n ˆ GOn(q) (it should be noted here though that Φnf (p) divides q +1, and hence does divide |G|). ∗ Also, Φnf (p) does not divide the order of GOn(q). Hence this case does not occur. group order m2 2 4 2m GO2m+1(q) 2q (q − 1)(q − 1) ··· (q − 1) + m(m−1) 2 4 2m−2 m GO2m(q) 2q (q − 1)(q − 1) ··· (q − 1)(q − 1) − m(m−1) 2 4 2m−2 m GO2m(q) 2q (q − 1)(q − 1) ··· (q − 1)(q + 1) Table 1: Orders of orthogonal groups. The reducible cases are excluded by the following, Lemma 3.5. f Lemma 3.5. Let q = p , let D = (V, B) be a block-transitive 2-(n,k,λ)q design, and let G be ∼ m(n−m) the stabiliser of the B inside ΓLn(q), and let Gˆ = G ∩ GLn(q). Then Gˆ 6 H = q · (GLm(q) × GLn−m(q)) for some m such that 0 8 ∗ i Proof. By definition, Φnf (p) is coprime to each factor q − 1 dividing |H|, that is, for i 6 ∗ n ∗ max{m, n − m} < n. Since Φnf (p) divides q − 1, it follows that Φnf (p) is also coprime to qm(n−m), and hence does not divide the order of Gˆ. 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, Fn f where V = q , and let q = p , where p is prime. By Lemma 1.5, we may assume that t =2, and hence that k > 3. By Lemma 1.4, we may assume that k 6 n/2, so that n > 6. Let G be the setwise stabiliser of B inside ΓLn(q). We may then apply Lemma 3.2. For part 1 of Lemma 3.2, ∗ e 6 Zsigmondy’s theorem [30] states that if Φe(p)=1 then p =2 . However, Lemmas 2.3 and 2.4 n n−1 6 rule out the existence of any 2-(n,k,λ)q design with q or q equal to 2 . Lemma 3.3 rules out the occurrence of part 2 of Lemma 3.2. This leaves the possibility that part 3 of Lemma 3.2 ∗ ∗ ˆ ˆ holds, that is, Φnf (p) and Φ(n−1)f (p) divide the order of G = G ∩ GLn(q), and G is as in one of 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 ruled out by Lemma 3.5. The remaining imprimitive, symplectic type, and nearly simple cases are ∗ ∗ immediately excluded by divisibility conditions, since non-trivial Φnf (p) and Φ(n−1)f (p) are re- quired to divide the order of G ∩ GLn(q); see Remark 3.6 for more detail. This completes the proof 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 > 6 (d in the tables of [3]) and that the variable e in [3] must be able to take both the values nf and (n−1)f for the same group action. In particular, many of the groups in the tables of [3] only allow e to take one specific value in a given group action, allowing such a group to be immediately ruled out. The reader should also note that the meaning of the variable n as used in [3] for the alternating group case is not the same as the meaning of n here. 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