Up to isomorphism there are four symmetric (36,15,6) designs with automorphisms of order 7. Full automorphism group of one of them is the Chevalley group G(2,2) = U(3,3) : Z2 of order 12096. Unitary group U(3,3) acts transitively on that design

Dean Crnković

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GLASNIK MATEMATICKIVol. 34(54)(1999), 1 - 3SYMMETRIC (36,15,6) DESIGN HAVING U(3,3) AS ANAUTOMORPHISM GROUPDEAN CRNKOVICABSTRACT. Up to isomorphism there are four symmetric (36,15,6) de-signs with automorphisms of order 7. Full automorphism group of oneof them is the Chevalley group G(2, 2) ~ U(3,3) : Z2 of order 12096.Unitary group U(3,3) acts transitively on that design.1. INTRODUCTION AND PRELIMINARIESA symmetric (v, k, >') design is a finite incidence structure (P, B, I), whereP and E are disjoint sets and I ~ P x E, with the following properties:1. IPI = lEI = v,2. Every element of E is incident with exactly k elements of P,3. Every pair of elements of P is incident with exactly>' elements of E.Let V = (P, B,I) be a symmetric (v, k, >') design and G ~ AutV. GroupG has the same number of point and block orbits. Let us denote the numberof G-orbits by t, point orbits by PI,'" ,Pt, block orbits by El, ... ,Et, andput IPrl = Wr, IEil = ni. We shall denote points of the orbit Pr by Pr ={rl, ... ,r Wr - d. Further, denote by "fir the number of points of P r which areincident with the representative of the block orbit Ei. For those numbers thefollowing equalities hold:(1)tL"firr=lk,(2)Definition 1. The (t x t) -matrix bir) with entries satisfying properties (1)and (2) is called the orbit structure for parameters (v, k, >') and orbit distri-bution (WI,'" ,Wt), (nl, ... ,nt).1991 Mathematics Subject Classification. 05B05.Key words and phrases. symmetric design, automorphism group, orbit structure.2 DEAN CRNKOVICDefinition 2. The set of indices of points of the orbit Pr indicating whichpoints of Pr are incident with the representative of the block orbit Bi is calledthe index set for the position (i, r) of the orbit structure.2. CONSTRUCTION OF THE DESIGNLet p be an automorphism of a symmetric design. We shall denote by F(p)the number of points fixed by p. In that case, the number of blocks fixed byp is also F(p).Lemma 1. Let p be an automorphism of a symmetric (36,15,6) design. IfIpl = 7, then F(p) = 1.Proof It is known that F(p) < k+.,fii and F(p) == v(mod Ipl). Therefore,F(p) E {l, 8, 15}. If F(p) = 8, then the fixed structure must be a symmetric(8,8,6) design. Such a design doesn't exist, therefore F(p) t- 8. The caseF(p) = 15 can be eliminated in the similar way. 0Lemma 2. Up to isomorphism there are exactly two orbit structures for cyclicautomorphism group of order 7 and a symmetric (36,15,6) design. Thosestructures are:OSI17 2110 17413 7052ProofSolving equations (1) and (2).0Theorem 3. Up to isomorphism there are four symmetric (36,15,6) designswith automorphism of order 7. Let us denote them by VI, V2, V3 and V4•Full automorphism groups of those designs are: AutVI ~ AutV2 ~ Frob21,AutV3 ~ G(2, 2), AutV4 ~ Frob21 x Z2'Proof Indexing of the column and row correponding to the fixed pointand block is trivial. Therefore, we shall take into consideration only right-lower (5 x 5) submatrices of orbit structures. Indexing of the structure OSIleads to designs VI, V2 and V3. Orbit structure OS2 leads to the design V4.Index sets which could occure in the case of OSI are:O={O}, ... , 6={6}, 7={O,I}, ... , 27={5,6}28 = {O, 1, 2}, , 62 = {4, 5, 6},63 = {O, 1,2,3}, , 97 = {3,4,5,6}SYMMETRIC (36,15,6) DESIGN HAVING...98= {0,1,2,3,4}, ... , 118= {2,3,4,5,6}.Design V3 is presented in terms of index sets as follows:364 0o 8131 3631 4431 5431 3136 44108 2424 9813 1131541311109With the help of the computer program by V. Tonchev, we got followigorders of full automorphism groups: IAutV11 = IAutV21 = 21, IAutV3112096, IAutV4I = 42. Using the GAP [5J we have determine that AutV3 ~G(2,2) and AutV4 ~ Frob21 x Z2. 0Derived Chevalley group G(2, 2)' is isomorphic to the unitary group U(3, 3)of order 6048. Simple group U(3,3) acts transitively on the design V3.We have also found out that automorphism groups Frob21 and Frob14acts on the design V3 with orbit distributions (1,7,7,21) and (1,7,7,7,14)respectively. It is interesting that U(3, 3) doesn't contain subgroup isomorphicto Frob14•It is obvious that the design V3 have null polarity. Therefore, it is possibleto construct strongly regular graph corresponding to that design.REFERENCES[1] M. Aschbacher, On Collineation Groups of Symmetric Block Designs, J. Combin. The-ory 11 (1971), 272-281.[2] F.C. Bussemaker, W.H. Haemers, J.J. Siedel, E. Spence, On (v, k,'x) Graphs andDesigns with Trivial Automorphism Groups, J. Combin. Theory, Series A 50 (1989),33-46.[3] J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker and R.A. Wilson, Atlas of FiniteGroups, Oxford, (1985).[4] V. Cepulic, On symmetric block designs (40,13,4) with automorphisms of order 5,Discrete Math. 128 (1994) no. 1-3,45-60.[5] GAP, Lehrstuhl D fuer Mathematik, RWTH Aachen.[6] Z. Janko, Coset Enumeration in Groups and Constructions of Symmetric Designs,Combinatorics '90, (1992), 275-277.[7] E. Lander, Symmetric Designs: An Algebraic Approach, Cambridge University Press(1983).[8J M.-O. Pavcevic, Symmetric designs of Menon series admitting an action of Frobeniusgroups, Glasnik Matematicki, (1996), 209-223.[9J W. D. Wallis, A. P. Street and J. S. Wallis, Combinatorics: Room Squares, Sum-FreeSets, Hadamard matrices, Springer Verlag, Berlin-Heidelberg-New York (1972).Address: ODSJEK ZA MATEMATIKU, FILOZOFSKI FAKULTET U RlJEel, OMLADINSKA 14, 51000RIJEKA, CROATIA(Received: 19.3.1999.)

Symmetric (36,15,6) design having U(3,3) as an automorphism group

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