We have proved that the first known symmetric (66,26,10) design, constructed by Tran van Trung, is up to isomorphism the only symmetric (66,26,10) design admitting an automorphism
of order 55. A full automorphism group of that design is isomorphic to 
Frob_{55}times D_{10}

D. Crnković

S. Rukavina

English

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Mathematical Communications 6(2001), 83-87 83Unique symmetric (66,26,10) design admitting anautomorphism of order 55Dean Crnković∗and Sanja Rukavina†Abstract.We have proved that the first known symmetric (66,26,10)design, constructed by Tran van Trung, is up to isomorphism the onlysymmetric (66,26,10) design admitting an automorphism of order 55. Afull automorphism group of that design is isomorphic to Frob55 ×D10.Key words: symmetric design, automorphism group, orbit structureAMS subject classifications: 05B05Received May 8, 2000 Accepted April 26, 20011. Introduction and preliminariesA symmetric (v, k, λ) design is a finite incidence structure (P ,B, I), where P andB are disjoint sets and I ⊆ P × B, with the following properties:1. |P| = |B| = v,2. every element of B is incident with exactly k elements of P ,3. every pair of elements of P is incident with exactly λ elements of B.Elements of the set P are called points and elements of the set B are called blocks.Let D = (P ,B, I) be a symmetric (v, k, λ) design and G ≤ AutD. Group Ghas the same number of point and block orbits. Let us denote the number ofG−orbits by t, point orbits by P1, . . . ,Pt, block orbits by B1, . . . ,Bt, and put|Pr| = ωr, |Bi| = Ωi. We shall denote points of the orbit Pr by r0, . . . , rωr−1, (i.e.Pr = {r0, . . . , rωr−1}). Further, denote by γir the number of points of Pr whichare incident with the representative of the block orbit Bi. For those numbers thefollowing equalities hold:t∑r=1γir = k , (1)t∑r=1Ωjωrγirγjr = λΩj + δij · (k − λ) . (2)∗Department of Mathematics, Faculty of Philosophy, Omladinska 14, HR-51 000 Rijeka, Croa-tia, e-mail: deanc@mapef.pefri.hr†Department of Mathematics, Faculty of Philosophy, Omladinska 14, HR-51 000 Rijeka, Croa-tia, e-mail: sanjar@mapef.pefri.hr84 D.Crnković and S.RukavinaDefinition 1. The (t × t)-matrix (γir) with entries satisfying properties (1)and (2) is called the orbit structure for parameters (v, k, λ) and orbit distribution(ω1, . . . , ωt), (Ω1, . . . ,Ωt).The first step of the construction of designs is to find all orbit structures (γir)for some parameters and orbit distribution. The next step, called indexing, is todetermine for each number γir exactly which points from the point orbit Pr areincident with the representative of the block orbit Bi. Because of a large number ofpossibilities, it is often necessary to involve a computer in both construction steps.Definition 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 called theindex set for the position (i, r) of the orbit structure.The first symmetric (66, 26, 10) design is constructed by Tran van Trung (see[8]). A full automorphism group of that design is isomorphic to Frob55 × D10. I.Matulić-Bedenić, K. Horvatić-Baldasar and E. Kramer (see [6]) have constructed18 mutually nonisomorphic symmetric (66, 26, 10) designs with Frobenius auto-morphism group of order 39. Later on, M.-O. Pavčević and E. Spence (see [7])have proved that there are up to isomorphism 558 symmetric (66, 26, 10) designsadmitting a dihedral automorphism group of order 10 acting with orbit distribu-tion (1, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5) on the sets of points and blocks, and 22 symmetric(66, 26, 10) designs having an elementary abelian automorphism group of order 25.Finally, it has been proved (see [2]) that there are up to isomorphism three sym-metric (66, 26, 10) designs with automorphism group isomorphic to Frob55. Amongconstructed symmetric (66, 26, 10) designs there are 590 mutually nonisomorphicones.Our aim is to construct symmetric (66, 26, 10) designs admitting an abelianautomorphism group of order 55, which will complete the classification of symmetric(66, 26, 10) designs with an automorphism group of order 55.2. Z55 acting on a symmetric (66,26,10) designFrom now on by G we shall denote a cyclic group of order 55 presented as follows:G ∼= 〈ρ, σ | ρ11 = 1, σ5 = 1, ρσ = 1〉.Let α be an automorphism of a symmetric design. By F (α) we shall denote thenumber of points fixed by α. In that case, the number of blocks fixed by α is alsoF (α).Lemma 1. Let ρ be an automorphism of a symmetric (66, 26, 10) design. If|ρ| = 11, then F (ρ) = 0.Proof. It is known that F (ρ) < k+√k − λ and F (ρ) ≡ v(mod |ρ|). Therefore,F (ρ) ∈ {0, 11, 22}. One cannot construct fixed blocks for F (ρ) ∈ {11, 22}. ✷Lemma 2. Let G be a cyclic automorphism group of order 55 of a symmetric(66, 26, 10) design D. G acts semistandardly on D with orbit distribution (11, 55).Proof. Group 〈ρ〉 acts on D with orbit distribution (11, 11, 11, 11, 11, 11). Since〈ρ〉  G, σ maps 〈ρ〉-orbits on 〈ρ〉-orbits. Therefore, only possibilities for orbitdistributions for the group G are (11, 11, 11, 11, 11, 11) and (11, 55). In the case oforbit distribution (11, 11, 11, 11, 11, 11), automorphism σ would act on D with 66Z55 acting on 2-(66,26,10) design 85fixed points (blocks), which is not possible. For orbit distribution (11, 55) one canconstruct only one orbit structure, namely:OS 11 5511 1 2555 5 21✷3. Construction of the designIt would be difficult to proceed with indexing of orbit structure OS. For example,there are(5525)possibilities for index sets for the position (1, 2) in the OS. There-fore, we shall use a principal series 〈1〉  〈ρ〉 G of the automorphism group G. Ouraim is to find all orbit structures for the group 〈ρ〉 corresponding to the structureOS. We shall construct designs from those orbit structures for 〈ρ〉, having in mindthe action of the permutation σ on 〈ρ〉-orbits.Theorem 1. Up to isomorphism there is only one symmetric (66, 26, 10) designwith an automorphism of order 55. Full automorphism group of that design isisomorphic to Frob55 ×D10 and its 2-rank is 31.Proof. The only orbit structure for the group 〈ρ〉 corresponding to the orbitstructure OS isOS1 11 11 11 11 11 1111 1 5 5 5 5 511 5 5 5 5 5 111 5 5 5 5 1 511 5 5 5 1 5 511 5 5 1 5 5 511 5 1 5 5 5 5We shall proceed with indexing of the orbit structure OS1, knowing that σ actson the set of six 〈ρ〉-orbits of blocks as permutation (1)(2, 3, 4, 5, 6), and on the setof point orbits as (1)(2, 6, 5, 4, 3). It is sufficient to determine index sets for the firstand second row of orbit structure OS1. Also, in the first row we have to determineindex sets only for positions (1, 1) and (1, 2). We shall assume that automorphismsρ and σ act on the set of points as follows:ρ = (I0, I1, . . . , I10), I = 1, 2, . . . , 6,σ = (1i)(2i, 6i, 5i, 4i, 3i), i = 0, 1, . . . , 10.Because of that action, index sets for positions (1, 2) and (2, 1) have to be basicblocks for symmetric (11, 5, 2) designs. As representatives of the first and secondblock orbit we shall choose the first blocks in those orbits, with respect to thelexicographical order. Index sets which could occur in designs are:0 = {0}, . . . , 10 = {10}, 11 = {0, 1, 2, 3, 4}, . . . , 472 = {6, 7, 8, 9, 10}.To eliminate isomorphic structures during indexing, we have used automor-phisms of the orbit structure OS1 which commute with permutation representation86 D.Crnković and S.Rukavinaof σ on the set of 〈ρ〉-orbits, and permutations αl, β1,k, β2,k ∈ NS(G), 2 ≤ l ≤10, 1 ≤ k ≤ 10, which act on the set of points as follows:riαl = rj , j ≡ (i · l)(mod 11), for r = 1, . . . , 6, i = 0, 1, . . . , 10,1iβ1,k = 1j , j ≡ (i+ k)(mod 11), for i = 0, 1, . . . , 10,riβ1,k = ri, for r = 2, . . . , 6, i = 0, 1, . . . , 10,1iβ2,k = 1i, for i = 0, 1, . . . , 10,riβ2,k = rj , j ≡ (i+ k)(mod 11), for r = 2, . . . , 6, i = 0, 1, . . . , 10.We have constructed up to isomorphism only one design, which is presented interms of index sets as follows:0 280 280 280 280 28020 20 450 450 20 520 450 450 20 5 2020 450 20 5 20 45020 20 5 20 450 45020 5 20 450 450 20Group Z55 is a subgroup of the group Frob55 ×D10. Therefore, it is clear thatthe constructed design is isomorphic to the one constructed by Tran van Trung withfull automorphism group isomorphic to Frob55 ×D10. By means of the computerprogram developed by V. Tonchev we have computed that 2-rank of the design is31. The constructed design is self-dual. ✷In [2] we have proved the following theorem:Theorem 2. Up to isomorphism there are three symmetric (66, 26, 10) designswith automorphism group Frob55. Let us denote them by D1, D2 and D3. Fullautomorphism groups of designs D1 and D2 are isomorphic to Frob55, and fullautomorphism group of the design D3 is isomorphic to Frob55 ×D10. Designs D1and D2 are dually isomorphic. 2−ranks of these three designs are 31.Remark 1. Design D3 is isomorphic to the design constructed by Tran vanTrung.Thereby we have proved the following statement:Theorem 3. Up to isomorphism there are three symmetric (66, 26, 10) designswith an automorphism group of order 55, namely the designs from Theorem2.References[1] M.Aschbacher, On Collineation Groups of Symmetric Block Designs, J.Combin. Theory 11(1971), 272–281.[2] D.Crnković, S.Rukavina, Symmetric (66,26,10) designs having Frob55 asan automorphism group, Glas. Mat., III. Ser. Vol. 35(55) (2000), 271-281.[3] V. Ćepulić, On symmetric block designs (40,13,4) with automorphisms of or-der 5, Discrete Math. 128(1994), 45–60.Z55 acting on 2-(66,26,10) design 87[4] Z. Janko, Coset Enumeration in Groups and Constructions of Symmetric De-signs, Combinatorics 90(1992), 275–277.[5] E. Lander, Symmetric Designs: An Algebraic Approach, Cambridge Univer-sity Press, 1983.[6] I.Matulić-Bedenić, K.Horvatić-Baldasar, E.Kramer, Constructionof new symmetric designs with parameters (66,26,10), J. Comb. Des. 3(1995),405–410.[7] M.-O. Pavčević, E. Spence, Some new symmetric designs with λ = 10 hav-ing an automorphism of order 5, Discrete Math. 196(1999), 257–266.[8] Tran van Trung, The existence of symmetric block designs with parameters(41,16,6) and (66,26,10), J. Comb. Theory A 33(1982), 201–204.

Unique symmetric (66,26,10) design admitting an automorphism of order 55

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