Discrete Mathematics 342 (2019) 2886–2892 Contents lists available at ScienceDirect Discrete Mathematics journal homepage: www.elsevier.com/locate/disc Automorphism groups of designs with λ D 1 ∗ William M. Kantor University of Oregon, Eugene, OR 97403, United States Northeastern University, Boston, MA 02115, United States article info a b s t r a c t Article history: If G is a finite group and k D q > 2 or k D q C 1 for a prime power q then, for infinitely Received 4 April 2018 many integers v, there is a 2-(v; k; 1)-design D for which AutD D∼ G. Accepted 30 January 2019 ' 2019 Elsevier B.V. All rights reserved. Available online 9 March 2019 Keywords: Automorphism group Design 1. Introduction Starting with Frucht's theorem on graphs [7], there have been many papers proving that any finite group is isomorphic to the full automorphism group of some specific type of combinatorial object. Babai surveyed this topic [3], and in [3, p. 8] stated that in [1] he had proved that 2-designs with λ D 1 are such objects when k D q > 2 or k D qC1 for a prime power q. (The case of Steiner triple systems was handled in [13].) The purpose of this note is to provide a proof of Babai's result1: Theorem 1.1. Let G be a finite group and q a prime power. (i) There are infinitely many integers v such that there is a 2-(v; q C 1; 1)-design D for which AutD D∼ G. (ii) If q > 2 then there are infinitely many integers v such that there is a 2-(v; q; 1)-design D for which AutD D∼ G. Parts of our proof mimic [5, Sec. 5] and [9, Sec. 4], but the present situation is much simpler. We modify a small number of subspaces of a projective or affine space in such a way that the projective or affine space can be recovered from the resulting design by elementary geometric arguments. Further geometric arguments determine the automorphism group. Section7 contains further properties of the design D in the theorem, some of which are needed in future research [6]. Notation: We use standard permutation group notation, such as xπ for the image of a point x under a permutation π and gh D h−1gh for conjugation. The group of automorphisms of a projective space Y D PG(V ) defined by a vector space V is denoted by P0L(V ) D P0L(Y ); this is induced by the group 0L(V ) of invertible semilinear transformations on V . Also A0L(V ) denotes the group of automorphisms of the affine space AG(V ) defined by V . 2. A simple projective construction Let G be a finite group. Let Γ be a simple, undirected, connected graph on f1;:::; ng such that AutΓ D∼ G and G acts semiregularly on the vertices. There is such a graph for each n ≥ 6jGj that is a multiple of jGj (using [2]). ∗ Correspondence to: University of Oregon, Eugene, OR 97403, United States. E-mail address: [email protected]. 1 This theorem was proved before I knew of Babai's result. https://doi.org/10.1016/j.disc.2019.01.030 0012-365X/' 2019 Elsevier B.V. All rights reserved. W.M. Kantor / Discrete Mathematics 342 (2019) 2886–2892 2887 ∗ Let K D Fq ⊂ F D Fq4 , and let θ generate F . Let VF be an n-dimensional vector space over F, with basis v1; : : : ; vn. View G as acting on VF , permuting fv1; : : : ; vng as it does f1;:::; ng. View VF as a vector space V over K. If Y is a set of points of P D PG(V ) then hY i denotes the smallest subspace of P containing Y . We will modify the point-line design PG1(V ) of P, using nonisomorphic designs ∆1 and ∆2 whose parameters are those 4 of PG1(K ) D PG1(3; q) but are not isomorphic to that design, chosen so that Aut∆1 fixes a point (Proposition 3.5). Our design D has the set P of points of P as its set of points. Most blocks of D are lines of P, with the following exceptions involving some of the subspaces Fv, 0 6D v 2 V , viewed as subsets of P. For orbit representatives i and ij of G on the vertices and ordered edges of Γ , (I) replace the set of lines of PG1(Fvi) by a copy of the set of blocks of ∆1, subject only to the condition (#) there are distinct blocks, neither of which is a line of P, whose span in P is PG1(Fvi), g and then apply all g 2 G to these sets of blocks in order to obtain the blocks in PG1((Fvi) ), g 2 GI and (II) replace the set of lines of PG1(F(vi C θvj)) by a copy of the set of blocks of ∆2, subject only to (#), and then apply all g g 2 G to these sets of blocks in order to obtain the blocks in PG1(F(vi C θvj) ), g 2 G. We need to check that these requirements can be met. N (i) Satisfying (#): Let ∆s be an isomorphic copy of ∆s, s D 1 or 2, whose set of points is that of PG1(Fv) D PG1(Fvi) or N PG1(F(vi C θvj)). Let B1 and B2 be any distinct blocks of ∆s. Choose any permutation π of the points of PG1(Fv) such that the π π N π N C ≥ sets B1 and B2 are not lines of PG1(Fv) and together span PG1(Fv). Using ∆s in place of ∆s satisfies (#). (If q 1 4 then B2 is not needed.) 0 g g 0 (ii) These replacements are well-defined: For (II), if F(vi C θvj) \ F(vi C θvj) 6D 0 for some g; g 2 G, then v g0 C θv g0 2 ∗ i j F(vig C θvjg ). Then either v g0 D vig and v g0 D vjg , or v g0 D αθvjg and θv g0 D αvig for some α 2 F ; but in the latter case i j ∗ i j D D 0 D g f g we obtain 1 αθ and θ α, whereas θ generates F . Thus, vig vi , so the semiregularity of G on 1;:::; n implies that g0 D g, as required. It is trivial to see that D is a design having the same parameters as PG1(V ). Clearly G acts on the collection of subsets of P occurring in (I) or (II): we can view G as a subgroup of both AutD and PGL(V ). We emphasize that the sets in (I) and (II) occupy a tiny portion of the underlying projective space: most sets Fv are unchanged. More precisely, in view of the definition of D: Every block of D not contained in a set (I) or (II) is a line of P: (2.1) Every line of P not contained in set (I) or (II) is a block of D: Nevertheless, we will distinguish between the lines of P and the blocks of D, even when the blocks happen to be lines. A subspace of D is a set of points that contains the block joining any pair of its points. (Examples: (I) and (II) involve subspaces of D.) A hyperplane of D is a subspace of D that meets every block but does not contain every point. We need further notation: Distinct y; z 2 P determine a block yz of D and a linehy; zi of P: (2.2) For distinct y; z 2 P and x 2 P − yz; (2.3) hxjy; zi D S{xp j p 2 y0z0; y0 2 xy − fxg; z0 2 xz − fxg; fy; zg 6D fy0; z0g}: Here(2.3) depends only on D not on P, which will allow us to recover P from D. 2 P 1 jPj 2 P − Lemma 2.4. If y; z are distinct, then there are more than 2 points x yz such that (1) hx; y; zi is a plane of P every line of which, except possibly hy; zi, is a block of D, (2) hxjy; zi D hx; y; zi, (3) if yz ⊆ hxjy; zi then hy; zi D yz, and (4) if yz 6⊆ hxjy; zi then hy; zi is the union of the pairs fy1; z1g ⊂ hxjy; zi such that y1z1 6⊆ hxjy; zi. Proof. Let [ x 2= yz [ {hy; z; Fvi j Fv in (I) or (II) }: (2.5) 4n − − − 2 6 − − − C 1 jPj h i There are more than (q 1)=(q 1) n (q 1)=(q 1) (q 1) > 2 such points x. Clearly x; y; z is a plane of P. (1) Let L 6D hy; zi be a line of hx; y; zi, so hx; y; zi D hy; z; Li. If L is not a block of D then, by(2.1), L is contained in some set Fv in (I) or (II), so x 2 hy; z; Li ⊆ hy; z; Fvi contradicts(2.5). (2) By (1), hx; yi and hx; zi are blocks of D. Let fy0; z0g be as in(2.3). Then fy0; z0g ⊂ hx; y; zi and hy0; z0i 6D hy; zi. By (1), y0z0 D hy0; z0i ⊆ hx; y; zi and xp D hx; pi ⊆ hx; y; zi for each point p of hy0; z0i. Then hxjy; zi ⊆ hx; y; zi. Each point of hx; y; zi lies in such a line hx; pi; since that line is a block by (1), hx; y; zi ⊆ hxjy; zi.