Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-05T13:49:06.509Z Has data issue: false hasContentIssue false
  • English
  • Français

Large doubly transitive orbits on a line

Part of: Permutation groups Finite geometry and special incidence structures

Published online by Cambridge University Press:  09 April 2009

Alessandro Montinaro
Alessandro Montinaro
Affiliation:
Dipartimento di Matematica, Universitá degli Studi di LecceVia per Arnesano73100 LecceItalye-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Projective planes of order n with a coUineation group admitting a 2-transitive orbit on a line of length at least n/2 are investigated and new examples are provided.

MSC classification

Secondary: 51E15: Affine and projective planes 20B25: Finite automorphism groups of algebraic, geometric, or combinatorial structures
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 83 , Issue 2 , October 2007 , pp. 227 - 270
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Aschbacher, M., Finite group theory (Cambridge University Press, 1996).Google Scholar
[2]Bender, H., ‘Endliche zweifach transitive permutationsgruppen, deren involutionen keine fixpunkte haben’, Math. Z. 104 (1968), 175–204.CrossRefGoogle Scholar
[3]Bender, H., ‘Transitive gruppen gerader ordnung, in denen jede involution genau einen puntk festläß t, J. Algebra 17 (1971), 527–554.CrossRefGoogle Scholar
[4]Biliotti, M. and Francot, E., ‘Two-transitive orbits in finite projective planes’, J Geom. 82 (2005), 1–24.CrossRefGoogle Scholar
[5]Biliotti, M., Jha, V. and Johnson, N. L., ‘The collineation group of generalized twisted fields planes’, Geom. Dedicata 76 (1999), 97–126.CrossRefGoogle Scholar
[6]Biliotti, M. and Johnson, N. L., ‘The non-solvable rank 3 affine planes’, J Combin. Theory Ser. A 93 (2000), 201–230.CrossRefGoogle Scholar
[7]Biliotti, M. and Korchmáros, G., ‘Some new results on collineation groups preserving an oval of a finite projective plane’, in: Combinatorics ' 88, Vol. I (Ravello, 1988), Res. Lecture Notes Math., Mediterranean, Rende (1991) pp. 159–170.Google Scholar
[8]Biliotti, M. and Montinaro, A., ‘Finite projective planes of order n with a 2-transitive orbit of length n – 3’, Adv. Geom. 6 (2005), 15–37.CrossRefGoogle Scholar
[9]Cofman, J., ‘Double transitivity in finite affine and projective planes’, Atti Accad. Naz, Lincei, Rend. Cl. Sci. Fis. Mat. Nat. 8 (1967), 317–320.Google Scholar
[10]Cofman, J., ‘On a conjecture of Hughes’, Proc. Camb. Phil. Soc. 63 (1967), 647–652.CrossRefGoogle Scholar
[11]Conway, J. H., Curtis, R. T., Parker, R. A. and Wilson, R. A., Atlas of Finite Croups. Maximal subgroups and ordinary charactersfor simple groups (Oxford University Press, 1985).Google Scholar
[12]Cooperstein, B. N., ‘Minimal degree for a permutation representation of a classical group’, Israel J. Math. 30 (1978), 213–235.CrossRefGoogle Scholar
[13]Czerwinski, T., ‘Finite translation planes with a collineation groups doubly transitive on the points at infinity’, J. Algebra 22 (1972), 428–441.CrossRefGoogle Scholar
[14]Czerwinski, T., ‘On collineation groups that fix a line of a finite projective plane’, Illinois J. Math. 16 (1977), 221–230.Google Scholar
[15]Dempwolff, U., ‘The projective planes of order 16 admitting SL.(3, 2)’, Rad. Mat. 7 (1991), 123–134.Google Scholar
[16]Dempwolff, U. and Reifart, A., ‘The classification of the translation planes of order 16. I’, Geom. Dedicata 15 (1984), 137–153.Google Scholar
[17]Dixon, J. D. and Mortimer, B., Permutation groups (Springer Verlag, New York, 1966).Google Scholar
[18]Foulser, D. A., ‘Solvable flag transitive affine groups’, Math. Z. 86 (1964), 191–204.CrossRefGoogle Scholar
[19]Ganley, M. J. and Jha, V., ‘On translation planes with a 2-transitive orbit on the line at infinity’, Arch. Math. (Basel) 47 (1986), 379–384.CrossRefGoogle Scholar
[20]Ganley, M. J., Jha, V. and Johnson, N. L., ‘The translation planes admitting a nonsolvable doubly transitive line-sized orbit’, J. Geom. 69 (2000), 88–109.CrossRefGoogle Scholar
[21]Glauberman, G., ‘Central elements in core-free groups’, J. Algebra 4 (1966), 403–420.CrossRefGoogle Scholar
[22]Gorenstein, D., Finite groups (Chelsea Publishing Company, New York, 1980).Google Scholar
[23]Gorenstein, D. and Walter, J. H., ‘The characterization of finite groups with dihedral Sylow 2-subgroups I’, J. Algebra 2 (1965), 85–151.CrossRefGoogle Scholar
[24] The GAP Group, ‘Gap – groups, algorithms, and programming, version 4.3’, http://www.gap-system.org, 2002.Google Scholar
[25]Hartley, R. W., ‘Determination of the ternary collineation groups whose coefficients lie in GF(2n)’, Ann. Math. 27 (1926), 140–158.CrossRefGoogle Scholar
[26]Hering, C., ‘Eine bemerkung über automorphismengruppen von endlichen projektiven ebenen und möbiusebenen’. Arch. Math. (Basel) 18 (1967), 107–110.CrossRefGoogle Scholar
[27]Hering, C., ‘On involutorial elations of projective planes’, Math. Z. 132 (1973), 91–97.CrossRefGoogle Scholar
[28]Hering, C., ‘Transitive linear groups and linear groups which contain irreducible subgroups of prime order’, Geom. Dedicata 2 (1974), 425–60.CrossRefGoogle Scholar
[29]Hiramine, Y., ‘On finite affine planes with a 2-transitive orbit on l∞,’, J. Algebra 162 (1993), 392–409.CrossRefGoogle Scholar
[30]Ho, C. Y., ‘Involutory collineations offiniteplanes’, Math. Z. 193 (1986), 235–240.CrossRefGoogle Scholar
[31]Hiramine, Y., ‘Projective planes of order 15 and other odd composite orders’, Geom. Dedicata 27 (1988), 49–64.Google Scholar
[32]Ho, C. Y. and Gonçalves, A., ‘On totally irregular simple collineation groups’, in: Advances infinite geometries and designs (Chelwood Gate 1990) (eds. Hirschfeld, J. W. P., Hughes, P. R. and Thas, J. A.), Oxford Sci. Publ. (Oxford Univ. Press, New York, 1991) pp. 177–193.Google Scholar
[33]Hughes, D. R. and Piper, F. C., Projective Planes (Springer Verlag, New York – Berlin, 1973).Google Scholar
[34]Huppert, B., Endliche Gruppen I (Springer Verlag, New York – Berlin, 1967).CrossRefGoogle Scholar
[35]Huppert, B. and Blackburn, N., Finite Groups III (Springer Verlag, Berlin – Heidelberg – New York, 1982).CrossRefGoogle Scholar
[36]Janko, Z. and Van Trung, T., ‘The full collineation group of any projective plane of order 12 is a {2, 3}-group’, Geom. Dedicata 12 (1982), 101–110.CrossRefGoogle Scholar
[37]Johnson, N. L., ‘A note on the derived semifield planes of order 16’, Aequationes Math. 18 (1978), 103–111.CrossRefGoogle Scholar
[38]Kallaher, M., ‘Translation planes’, in: Handbook Of Incidence Geometry (ed. Buekenhout, F.) (Elsevier, 1995) pp. 137–192.CrossRefGoogle Scholar
[39]Kantor, W. M., ‘On unitary polarities of finite projective planes’, Canad. J. Math. 23 (1971), 1060–1077.CrossRefGoogle Scholar
[40]Kantor, W. M., ‘Homogeneous designs and geometric lattices’, J. Combin. Theory Ser. A 38 (1985), 66–74.CrossRefGoogle Scholar
[41]Karpilovsky, G., The Schur Multiplier (Clarendon Press, Oxford, 1987).Google Scholar
[42]Kleidman, P. B., ‘The maximal subgroups of the chevalley groups G 2(q) with q odd, the Ree groups 2G 2(q), and their automorphism groups’, J. Algebra 117 (1988), 30–71.CrossRefGoogle Scholar
[43]Kleidman, P. B. and Liebeck, M., The subgroup structure of the finite classical groups (Cambridge University Press, Cambridge, 1990).CrossRefGoogle Scholar
[44]Korchmáros, G., ‘Collineation groups doubly transitive on the points at infinity in an affine plane of order 2r’, Arch. Math. (Basel) 37 (1981), 572–576.CrossRefGoogle Scholar
[45]Li, X., ‘A characterization of the finite simple groups’, J. Algebra 245 (2001), 620–649.Google Scholar
[46]Liebeck, M. W., ‘On the order of maximal subgroups of the finite classical groups’, Proc. London Math. Soc. (3) 50 (1985), 426–446.CrossRefGoogle Scholar
[47]Matulić-Bedenić, I., ‘The classification of projective planes oforder 11 which possess an involution’, Rod. Mat. 1 (1985), 149–157.Google Scholar
[48]Matulić-Bedenić, I., ‘The classification of projective planes of order 13 which possess an involution’, Rad Hrvatske Akad. Znam. Umjet. 456 (1991), 9–13.Google Scholar
[49]Mitchell, H. H., ‘Determination of ordinary and modular ternary linear groups’, Trans. Amer. Math. Soc. 12 (1911), 207–242.CrossRefGoogle Scholar
[50]Mwene, B., ‘On the subgroups of the group PSL4(2m)’, J. Algebra 41 (1976), 79–107.CrossRefGoogle Scholar
[51]Passman, D. S., Permutation Groups (W. A. Benjamin, Inc., New York -Amsterdam, 1968).Google Scholar
[52]Penttila, T., Royle, G. F. and Simpson, M. K., ‘Hyperovals in the known projective planes of order 16. II’, J. Combin. Des. 4 (1996), 59–65.3.0.CO;2-Z>CrossRefGoogle Scholar
[53]Reifart, A., ‘The classification of the translation planes of order 16. II’, Geom. Dedicata 17 (1984), 1–9.CrossRefGoogle Scholar
[54]Ribenboim, P., Catalan's conjecture (Acad. Press, Boston, 1994).Google Scholar
[55]Ribenboim, P., Fermat's last theorem for amateurs (Springer-Verlag, New York, 1999).Google Scholar
[56]Schulz, R. H., ‘Über translationsebenen mit kollineationsgruppen, die die punkte der ausgezeich-neten geraden zweinfach transitiv permutieren’, Math. Z. 122 (1971), 246–266.CrossRefGoogle Scholar
[57]Shull, R., ‘Collineations of projective planes of order 9’, J. Combin. Theory Ser. A 37 (1984), 99–120.CrossRefGoogle Scholar
[58]Shull, R., ‘The classification of projective planes of order 9 possessing a collineation group of order 5’, Algebras Groups Geom. 2 (1985), 365–379.Google Scholar
[59]Suzuki, M., ‘On a class of doubly transitive groups’, Ann. of Math. (2) 75 (1962), 105–145.CrossRefGoogle Scholar
[60]Yu, L. and Le, M., ‘On the diophantine equation (xn – 1)/(x – 1) = ym’, Ada Arith. 21 (1972), 299–301.Google Scholar