International Journal of Algebra, Vol. 10, 2016, no. 1, 27-32 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2016.51277 Block-Transitive 4 (v, k, 4) Designs and Suzuki Groups Shaojun Dai Department of Mathematics, Tianjin Polytechnic University No.399 Binshuixi Road, Xiqing District Tianjin, 300387, P. R. China Shangzhao Li School of Mathematics and Statistics, Changshu Institute of Technology Jiangsu, 215500 P. R. China Copyright c 2015 Shaojun Dai and Shangzhao Li. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract This article is a contribution to the study of the automorphism groups of 4 (v, k, λ) designs. Let S = (P, B) be a non-trivial 4 (q 2 + 1, k, 4) design, where q = 2 2n+1 for some positive integer n 1, and G Aut(S) acts block-transitively on S. If the socle of G is isomorphic to the simple groups of Lie type Sz(q), then G is not flag-transitive. Mathematics Subject Classification: 05B05, 20B25 Keywords: flag-transitive; block-transitive; t design; Suzuki group 1 Introduction Following [2], for positive integers t k v and λ, we define a t (v, k, λ) design to be a finite incidence structure S = (P, B). Here P denotes a set of points, P = v, and B a set of blocks, B = b, with the properties that each block is incident with k points, and each t subset of P is incident with λ blocks. A flag of S is an incident point-block pair (x, B) with x is incident with B, where B B. We consider automorphisms of S as pairs
28 Shaojun Dai and Shangzhao Li of permutations on P and B which preserve incidence structure. We call a group G Aut(S) of automorphisms of S flag-transitive(respectively blocktransitive, point t transitive, point t homogeneous) if G acts transitively on the flags(respectively transitively on the blocks, t transitively on the points, t homogeneously on the points) of S. For short, S is said to be, e.g., flagtransitive if S admits a flag-transitive group of automorphisms. For historical reasons, a t (v, k, λ) design with λ = 1 is called a Steiner t design (sometimes this is also known as a Steiner system). If t < k < v holds, then we speak of a non-trivial Steiner t designs. Investigating t designs for arbitrary λ, but large t, Cameron and Praeger proved the following result: Theorem 1.1 ([1]) Let S = (P, B) be a t (v, k, λ) design. If G Aut(S) acts block-transitively on S, then t 7, while if G Aut(S) acts flagtransitively on S, then t 6. Recently, Huber([2])completely classified all flag-transitive Steiner t designs using the classification of the finite 2 transitive permutation groups. Hence the determination of all flag-transitive and block-transitive t designs with λ 2 has remained of particular interest and has been known as a longstanding and still open problem. In 2010, Xu([3])completely classified flag-transitive 6 (v, k, λ) designs with λ 5. In 2010, Liu([4]) completely classified flag-transitive 5 (v, k, 2) design and P SL(2, q). In 2011, Xu([5])completely classified two classes Lie type simple groups and flag-transitive 4 (v, k, 2) designs. In this paper, we discussed the block-transitive 4 (v, k, 4) designs and Sz(q) and got the following result: Main Theorem. Let S = (P, B) be a non-trivial 4 (q 2 + 1, k, 4) design, where q = 2 2n+1 for some positive integer n 1, and G Aut(S) acts block-transitively on S. If Soc(G), the socle of G, is Sz(q), then G is not flag-transitive. The second section describes the definitions and contains several preliminary results about flag-transitivity and t designs. In the third section we give the proof of the Main Theorem. 2 Preliminary Results The Suzuki groups Sz(q) form an infinite family of simple groups of Lie type, and were defined in [6] and [7] as subgroups of SL(4, q). Let GF (q) be finite field of q elements, where q = 2 2n+1 for some positive integer n 1 (in particular, q 8). Let Q is a Sylow 2-subgroup of G, K is a multiplicative group of
Block-transitive 4 (v, k, 4) designs and Suzuki groups 29 GF (q) and Sz(q) is a group of order q 2 (q 2 + 1)(q 1)(see [8, 9]). Hence Sz(q) is a group of automorphisms of Steiner 3 (q 2 + 1, q + 1, 1) design and acts 2 transitive on q 2 + 1 points(see [10]). Here we gather notation which are used throughout this paper. For a t design S = (P, B) with G Aut(S), let r denotes the number of blocks through a given point, G x denotes the stabilizer of a point x P and G B the setwise stabilizer of a block B B. We define G xb = G x G B. For integers m and n, let(m, n) denote the greatest common divisor of m and n, and m n if m divides n. Lemma 2.1 ([2]) Let G act flag-transitively on t (v, k, λ) design S = (P, B). Then G is block-transitive and the following cases hold: (1) G = G x x G = G x v, where x P; (2) G = G B B G = G B b, where B B; (3) G = G xb (x, B) G = G xb bk, where x B. Lemma 2.2 ([11]) Let S = (P, B) is a non-trivial t (v, k, λ) design. Then λ(v t + 1) (k t + 2)(k t + 1). Lemma 2.3 ([11]) Let S = (P, B) is a non-trivial 4 (v, k, λ) design. Then (1) bk = vr; (2) b = λv(v 1)(v 2)(v 3) k(k 1)(k 2)(k 3). Corollary 2.4 Let S = (P, B) is a non-trivial 4 (v, k, 4) design. If v = q 2 + 1, Then k 5+ 16q 2 31. 2 Proof. By Lemma 2.2, we have 4(v 3) (k 2)(k 3). If v = q 2 + 1, then 4(q 2 2) (k 2)(k 3). Hence We get k 2 5k 4q 2 + 14 0. k 5 + 16q 2 31. 2 3 Proof of the Main Theorem Suppose that G acts flag-transitively on 4 (v, k, 4) design and v = q 2 +1. Then G is block-transitive and point-transitive. Since T = Sz(q) G Aut(T ), we may assume that G = T : α and G = T : (G α ) by Dedekind s theorem, where α : x x 2, x GF (q) and α is an automorphism of field
30 Shaojun Dai and Shangzhao Li GF (q). Let q = 2 f, f = 2n + 1 is odd, and α = m, then m f. Obviously, G = q 2 (q 2 + 1)(q 1)m. First, we will proof that if g G fixes three different points of P, then g must fix at least five points in P. Suppose that g G, F ix P (g) 3, x F ix P (g). Let P is a normal Sylow 2-subgroup of G x. Then P is transitive on P {x}. By v = q 2 + 1, we have P = P {x} = q 2. Hence P acts regularly on P {x}. There exists h P such that z = y h for all y, z P {x}. Since g G x, h P and P is a normal Sylow 2-subgroup of G x, we have h 1 ghg 1 P. On the other hand, z h 1 ghg 1 = y ghg 1 = y hg 1 = z g 1 = z. So h 1 ghg 1 = 1, that is gh = hg. Hence h C = C P (g). We get that C is transitive on F ix P (g) {x}. Hence F ix P (g) {x} C. By C P, we have F ix P (g) {x} P. Note that P = q 2 = 2 2f, so F ix P (g) {x} 2 2f. Hence F ix P (g) {x} 0 (mod 2). It follows that F ix P (g) 1 (mod 2). For all h C T (g), y F ix P (g), we have y hg = y gh = y h. So y h F ix P (g). This means that F ix P (g) is a block which is fixed by C T (g). By C C T (g) and C is transitive on F ix P (g) {x} for all x F ix P (g), we have C T (g) is transitive on F ix P (g). Thus F ix P (g) C T (g). On the other hand, C T (g) T. Therefore, F ix P (g) T. Obviously, 3 T. Hence F ix P (g) = 3. By F ix P (g) 1 (mod 2), we get F ix P (g) 5. This means that g must fix at least five points in P. Now, we can continue to prove our main theorem. Obviously, α fixes three points of P which are 0, 1,. Then α G 0,1,. Hence α must fix at least five points in P. Since G acts block-transitively on 4 (v, k, 4) design, we can find four blocks, let B 1, B 2, B 3 and B 4, containing four points which is fixed by α. If α exchange B 1, B 2, B 3 and B 4, then 2 α which is impossible. Thus α must fix B 1, B 2, B 3 and B 4. We have G α G 0B1 = G 0B2 = G 0B3 = G 0B4. Therefore T acts also flag-transitively on 4 (q 2 + 1, k, 4) design. We may assume G = T and G = q 2 (q 2 + 1)(q 1). Since G acts flag-transitively on 4 (q 2 + 1, k, 4) design, we get G x = G v = q2 (q 2 + 1)(q 1) q 2 + 1 = q 2 (q 1), by Lemma 2.1(1). Again by Lemma 2.3(2)and Lemma 2.1(3), Thus b = 4v(v 1)(v 2)(v 3) k(k 1)(k 2)(k 3) = v G x k G xb. G xb = (k 1)(k 2)(k 3) Gx 4(v 1)(v 2)(v 3) = (k 1)(k 2)(k 3)q2 (q 1) 4q 2 (q 2 1)(q 2 2) = (k 1)(k 2)(k 3) 4(q+1)(q 2 2).
Block-transitive 4 (v, k, 4) designs and Suzuki groups 31 By Lemma 2.2, Again by Corollary 2.1, 4 G xb (q + 1)(q 2 2) = (k 1)(k 2)(k 3) (k 1) 4(v 3) = 4(k 1)(q 2 2), 1 G xb k 1 q + 1 3 + 16q2 31 2(q + 1) < 2. Hence G xb = 1. It follows that 4(q + 1)(q 2 2) = (k 1)(k 2)(k 3). We have k 2(2q 2 + 2q 2 4q 1). By Lemma 2.1, k G. Thus k (2(2q 2 + 2q 2 4q 1), q 2 (q 2 + 1)(q 1). Obviously, we get (2(2q 2 + 2q 2 4q 1), q 2 (q 2 + 1)(q 1) = 2(2q 2 + 2q 2 4q 1, (q 2 +1)(q 1)) = 2(q 2, 5) since q = 2 2n+1. If q 2(mod 5), then k 2. This is impossible since k 4. Hence q 2(mod 5) and k = 5, 10. If k = 10, then 4(q + 1)(q 2 2) = 9 8 7. If k = 5, then 4(q + 1)(q 2 2) = 4 3 2. There are impossible since q 8. This completes the proof the Main Theorem. Acknowledgements. Supported by the National Natural Science Foundation of China(11271028, 11301377), the Funding Programme for outstanding youth of Tianjin and the Scientific Research Fund of Heilongjiang Provincial Education Department (12541829). References [1] P. J. Cameron, C. E. Praeger, Block-transitive t designs, II: large t, F. De Clerck, et al. (Eds), Chapter in Finite Geometry and Combinatorics, London Math. Soc. Lecture Note Series, Vol. 191, Cambridge Univ. Press, 1993, 103-120. http://dx.doi.org/10.1017/cbo9780511526336.012 [2] M. Huber, Flag-transitive Steiner Designs, Birkhäuser Basel, Berlin, Boston, 2009. http://dx.doi.org/10.1007/978-3-0346-0002-6 [3] Xianghong Xu and Weijun Liu, On flag-transitive 6 (v, k, λ) designs with λ 5, Ars Combin., 97 (2010), 507-510. [4] Weijun Liu, Qionghua Tan and Luozhong Gong, Flag-transitive 5 (v, k, 2) designs, J. Jiangsu Univ., 31 (2010), 612-615.
32 Shaojun Dai and Shangzhao Li [5] Xianghong Xu, Lina Zhao and Weijun Liu, Two classes Lie type simple groups and flag-transitive 4 (v, k, 2) designs, Journal of Zhejiang Unviersity, 38 (2011), 4-6. [6] M. Suzuki, A new type of simple group of finite order, Proc. Nat. Acad. Sci. U.S.A., 46 (1960), 868-870. http://dx.doi.org/10.1073/pnas.46.6.868 [7] M. Suzuki, On a class of doubly transitive groups, Ann. of Math., 75 (1962), 105-145. http://dx.doi.org/10.2307/1970423 [8] Weijun Liu, Suzuki groups and automorphisms of finite linear spaces, Discrete Math., 269 (2003), 181-190. http://dx.doi.org/10.1016/s0012-365x(02)00752-5 [9] Shaojun Dai and Kun Zhao, Block transitive 2 (v, 13, 1) designs and Suzuki groups, Ars Combin., 105 (2012), 369-373. [10] J.D. Dixon, B. Mortimer, Permutation Groups, Springer Verlag, Berlin, 1996. http://dx.doi.org/10.1007/978-1-4612-0731-3 [11] H. Shen, The Theory of Combinatorial Design, Shanghai Jiao Tong Univ. Press, 1990. Received: December 28, 2015; Published: January 30, 2016