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Block transitive resolutions of \(t\)-designs and Room rectangles. (English) Zbl 0883.05014

Authors’ abstract: A resolution of \(t\)-designs is a partition of the trivial design \(X\choose k\) of all \(k\)-subsets of a \(v\)-set \(X\) into \(t\)-\((v',k,\lambda)\) designs, where \(v'\leq v\). A resolution of \(t\)-designs with \(v=v'\) is also called a large set of \(t\)-designs. A Room rectangle \(R\), based on \(X\choose k\), is a rectangular array whose non-empty entries are \(k\)-sets. This array has the further property that taken together the rows form a resolution of \(t_1\)-designs, and the columns form a resolution of \(t_2\)-designs. A resolution of \(t\)-designs for \(X\choose k\) is said to admit \(G\) as a block transitive automorphism group if \(G\) is \(k\)-homogeneous on \(X\), and permutes the \(t\)-designs of the resolution among themselves. Some examples of block transitive resolutions of nontrivial \(t\)-designs, for \(t\geq 2\), are: (1) an \(M_{11}\)-invariant set of 3-\((10,4,1)\) designs, (2) an \(M_{12}\)-invariant set of 4-\((11,5,1)\) designs, (3) an \(M_{24}\)-invariant set of 2-\((21,5,1)\) designs, (4) a \(\text{P}\Gamma\text{L}_2(2^s)\)-invariant set of 3-\((2^s,4,1)\) designs (\(s=3\) or 5), (5) a \(\text{P}\Gamma\text{L}_2(32)\)-invariant set of 2-\((16,4,1)\) designs, and (6) a variety of \(\text{PSL}_2(q)\)-invariant sets of 2-designs with \(k=3\). We show that this is a complete list. In particular there are no block transitive large sets of \(t\)-designs. Moreover, if \(1\neq a<b<c\) are odd integers such that \(\text{gcd}(a,b)=1\) and \(ab\) divides \(c\), then we construct a block transitive Room rectangle based on the 3-subsets of a \((7^c+1)\)-set whose rows are the Steiner triple systems on \(7^a\) points, and whose columns are Steiner triple systems on \(7^b\) points.

MSC:

05B05 Combinatorial aspects of block designs
05B15 Orthogonal arrays, Latin squares, Room squares
05B07 Triple systems

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[1] Beth, T.; Jungnickel, D.; Lenz, H., Design Theory (1986), Cambridge University Press: Cambridge University Press Cambridge
[2] Breach, D. R.; Street, A. P., Partitioning sets of quadruples into designs II, JCMCC, 3, 41-48 (1988) · Zbl 0714.05009
[3] Burnside, W., Theory of Groups of Finite Order (1958), Dover: Dover New York · JFM 28.0118.03
[4] Cameron, P. J., Finite permutation groups and finite simple groups, Bull. London Math. Soc., 13, 1-22 (1981) · Zbl 0463.20003
[5] Chee, Y. M.; Colbourne, C. J.; Furino, S. C.; Kreher, D. L., Large sets of disjoint \(t\)-designs, Australas. J. Combin., 2, 111-120 (1990) · Zbl 0757.05018
[6] Chouinard, L. G., Partitions of the 4-sets of a 13-set into disjoint projective planes, Discrete Math., 45, 297-300 (1983) · Zbl 0509.05014
[7] Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. P.; Wilson, R. A., Atlas of Finite Groups (1985), Clarendon Press: Clarendon Press Oxford · Zbl 0568.20001
[8] Curtis, C. W.; Iwahori, N.; Kilmoyer, R., Hecke algebras and characters of parabolic type of finite groups with \((B,N)\)-pairs, IHES Publ. Math., 40, 81-116 (1972) · Zbl 0254.20004
[9] Dembowski, P., (Finite Geometries, Ergebnisse der Mathematik, Vol. 44 (1968), Springer: Springer Heidelberg) · Zbl 0159.50001
[10] Dickson, L. E., Linear groups with an Exposition of the Galois Field Theory (1958), Dover: Dover New York
[11] Kantor, W. M., \(k\)-homogeneous groups, Math. Z., 124, 261-265 (1972) · Zbl 0232.20003
[12] Khosrovshahi, G. G.; Ajoodani-Namini, S., Combining \(t\)-designs, J. Combin. Theory, 58, 26-34 (1991), Ser. A · Zbl 0735.05021
[13] Kramer, E. S.; Magliveras, S. S., A 57 × 57 × 57 Room-type design, Ars Combin., 9, 163-166 (1980) · Zbl 0439.05016
[14] Kramer, E. S.; Kreher, D. L.; Magliveras, S. S.; Mesner, D. M., Coherent room rectangles from permutation groups, Ars Combinatoria, 9, 101-111 (1981) · Zbl 0439.05015
[15] Kramer, E. S.; Kreher, D. L.; Magliveras, S. S.; Mesner, D. M., An assortment of Room-type designs, Ars Combin., 11, 9-29 (1981) · Zbl 0473.05017
[16] Kramer, E. S.; Kreher, D. L.; Mesner, D. M., Some crowded Room rectangles, Ars Combin., 11, 71-85 (1982) · Zbl 0497.05015
[17] Kramer, E. S.; Magliveras, S. S.; Stinson, D. R., Some small large sets of \(t\)-designs, Australas. J. Combin., 3, 191-205 (1991) · Zbl 0763.05009
[18] Kramer, E. S.; Magliveras, S. S.; O’Brien, E. A., Some new large sets of t-designs, Australas. J. Combin., 7, 189-193 (1993) · Zbl 0778.05015
[19] Kreher, D. L., An infinite family of (simple) 6-designs, J. Combin. Designs, 1, 4, 277-280 (1993) · Zbl 0886.05020
[20] Liebeck, M. W., The affine permutation groups of rank three, (Proc. London Math. Soc., 54(3) (1987)), 477-516 · Zbl 0621.20001
[21] Livingstone, D.; Wagner, A., Transitivity of finite permutation groups on unordered sets, Math. Z., 90, 393-403 (1965) · Zbl 0136.28101
[22] Magliveras, S. S.; Plambeck, T. E., New infinite families of simple 5-designs, J. Combin. Theory, 44, 1-5 (1987), Ser. A · Zbl 0612.05013
[23] Shary, M. J., Partitioning sets of quintuples into designs, JCMCC, 6, 67-103 (1989) · Zbl 0692.05010
[24] Sharry, M. J.; Street, A. P., Partitioning sets of blocks into designs II, Australas. J. Combin., 3, 111-140 (1991) · Zbl 0763.05012
[25] Sharry, M. J.; Street, A. P., Partitioning sets of quadruples into designs III, Discrete Math., 92, 341-359 (1991) · Zbl 0757.05021
[26] Teirlinck, L., Non-trivial \(t\)-designs without repeated blocks exist for all \(t\), Discrete Math., 65, 301-311 (1987) · Zbl 0625.05005
[27] Teirlinck, L., Large sets of disjoint designs and related structures, (Dinitz, J. H.; Stinson, D. R., Contemporary Design Theory: A Collection of Surveys (1992), Willey: Willey New York), 561-592 · Zbl 0805.05012
[28] Tran, van Trung, On the existence of an infinite family of simple 5-designs, Math. Z., 187, 285-287 (1984) · Zbl 0537.05007
[29] Tran, van Trung, On the construction of \(t\)-designs and the existence of some new infinite families of simple 5-designs, Arch. Math., 47, 187-192 (1986) · Zbl 0608.05010
[30] Tsaranov, S. V.; Komissartschik, E. A., Intersections of maximal subgroups in simple groups of order less than \(10^6\), Comm. Algebra, 14, 9, 1623-1678 (1986) · Zbl 0598.20013
[31] Wielandt, H., Finite Permutation Groups (1964), Academic Press: Academic Press New York · Zbl 0138.02501
[32] Wu, Qiu-rong, A note on extending \(t\)-designs, Australas. J. Combin., 4, 229-235 (1991) · Zbl 0763.05013
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