J. Aust. Math. Soc. 83 (2007), no. 2, pp. 227–269.
Large doubly transitive orbits on a line
Alessandro Montinaro
Dipartimento di Matematica
Università degli Studi di Lecce
Via per Arnesano
73100 Lecce
Italy
alessandro.montinaro@unile.it
Received 4 November 2005; revised 1 June 2006
Communicated by L. Batten

Abstract

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

Download the article in PDF format (size 480 Kb)

2000 Mathematics Subject Classification: primary 51E15; secondary 20B25
(Metadata: XML, RSS, BibTeX)

References

  1. M. Aschbacher, Finite group theory (Cambridge University Press, 1996). MR1777008
  2. H. Bender, ‘Endliche zweifach transitive permutationsgruppen, deren involutionen keine fixpunkte haben’, Math. Z. 104 (1968), 175–204. MR227261
  3. H. Bender, ‘Transitive gruppen gerader ordnung, in denen jede involution genau einen puntk festläßt’, J. Algebra 17 (1971), 527–554. MR288172
  4. M. Biliotti and E. Francot, ‘Two-transitive orbits in finite projective planes’, J. Geom. 82 (2005), 1–24. MR2161811
  5. M. Biliotti, V. Jha and N. L. Johnson, ‘The collineation group of generalized twisted fields planes’, Geom. Dedicata 76 (1999), 97–126. MR1699202
  6. M. Biliotti and N. L. Johnson, ‘The non-solvable rank 3 affine planes’, J. Combin. Theory Ser. A 93 (2000), 201–230. MR1805294
  7. M. Biliotti and G. Korchmáros, ‘Some new results on collineation groups preserving an oval of a finite projective plane’, in: Combinatorics '88, Vol. 1 (Ravello, 1988), Res. Lecture Notes Math., Mediterranean, Rende (1991) pp. 159–170. MR1223562
  8. M. Biliotti and A. Montinaro, ‘Finite projective planes of order n with a 2-transitive orbit of length n-3’, Adv. Geom. 6 (2005), 15–37. MR2242859
  9. J. Cofman, ‘Double transitivity in finite affine and projective planes’, Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat. 8 (1967), 317–320. MR236813
  10. J. Cofman, ‘On a conjecture of Hughes’, Proc. Camb. Phil. Soc. 63 (1967), 647–652. MR217691
  11. J. H. Conway, R. T. Curtis, R. A. Parker and R. A. Wilson, Atlas of Finite Groups. Maximal subgroups and ordinary characters for simple groups (Oxford University Press, 1985). MR827219
  12. B.N. Cooperstein, ‘Minimal degree for a permutation representation of a classical group’, Israel J. Math. 30 (1978), 213–235. MR506701
  13. T. Czerwinski, ‘Finite translation planes with a collineation groups doubly transitive on the points at infinity’, J. Algebra 22 (1972), 428–441. MR313933
  14. T. Czerwinski, ‘On collineation groups that fix a line of a finite projective plane’, Illinois J. Math. 16 (1977), 221–230. MR487766
  15. U. Dempwolff, ‘The projective planes of order 16 admitting \operatorname{SL}(3,2)’, Rad. Mat. 7 (1991), 123–134. MR1126890
  16. U. Dempwolff and A. Reifart, ‘The classification of the translation planes of order 16. I’, Geom. Dedicata 15 (1984), 137–153. MR737954
  17. J. D. Dixon and B. Mortimer, Permutation groups (Springer Verlag, New York, 1966). MR1409812
  18. D. A. Foulser, ‘Solvable flag transitive affine groups’, Math. Z. 86 (1964), 191–204. MR170956
  19. M. J. Ganley and V. Jha, ‘On translation planes with a 2-transitive orbit on the line at infinity’, Arch. Math. (Basel) 47 (1986), 379–384. MR866529
  20. M. J. Ganley, V. Jha and N. L. Johnson, ‘The translation planes admitting a nonsolvable doubly transitive line-sized orbit’, J. Geom. 69 (2000), 88–109. MR1800461
  21. G. Glauberman, ‘Central elements in core-free groups’, J. Algebra 4 (1966), 403–420. MR202822
  22. D. Gorenstein, Finite groups (Chelsea Publishing Company, New York, 1980). MR569209
  23. D. Gorenstein and J. H. Walter, ‘The characterization of finite groups with dihedral Sylow 2-subgroups I’, J. Algebra 2 (1965), 85–151. MR177032
  24. The GAP Group, ‘Gap — groups, algorithms, and programming, version 4.3’, http://www.gap-system.org, 2002.
  25. R. W. Hartley, ‘Determination of the ternary collineation groups whose coefficients lie in \operatorname{GF}(2^{n})’, Ann. Math. 27 (1926), 140–158.
  26. C. Hering, ‘Eine bemerkung über automorphismengruppen von endlichen projektiven ebenen und möbiusebenen’, Arch. Math. (Basel) 18 (1967), 107–110. MR210793
  27. C. Hering, ‘On involutorial elations of projective planes’, Math. Z. 132 (1973), 91–97. MR325738
  28. C. Hering, ‘Transitive linear groups and linear groups which contain irreducible subgroups of prime order’, Geom. Dedicata 2 (1974), 425–460. MR335659
  29. Y. Hiramine, ‘On finite affine planes with a 2-transitive orbit on l_{∞}’, J. Algebra 162 (1993), 392–409. MR1254783
  30. C. Y. Ho, ‘Involutory collineations of finite planes’, Math. Z. 193 (1986), 235–240. MR856151
  31. C. Y. Ho, ‘Projective planes of order 15 and other odd composite orders’, Geom. Dedicata 27 (1988), 49–64. MR950322
  32. C. Y. Ho and A. Gonçalves, ‘On totally irregular simple collineation groups’, in: Advances in finite geometries and designs (Chelwood Gate 1990) (eds. J. W. P. Hirschfeld, P. R. Hughes and J. A. Thas), Oxford Sci. Publ. (Oxford Univ. Press, New York, 1991) pp. 177–193. MR1138743
  33. D. R. Hughes and F. C. Piper, Projective Planes (Springer Verlag, New York - Berlin, 1973). MR333959
  34. B. Huppert, Endliche Gruppen I (Springer Verlag, New York - Berlin, 1967). MR224703
  35. B. Huppert and N. Blackburn, Finite Groups III (Springer Verlag, Berlin - Heidelberg - New York, 1982). MR662826
  36. Z. Janko and T. Van Trung, ‘The full collineation group of any projective plane of order 12 is a {2,3}-group’, Geom. Dedicata 12 (1982), 101–110. MR645042
  37. N. L. Johnson, ‘A note on the derived semifield planes of order 16’, Aequationes Math. 18 (1978), 103–111. MR513872
  38. M. Kallaher, ‘Translation planes’, in: Handbook Of Incidence Geometry (ed. F. Buekenhout) (Elsevier, 1995) pp. 137–192. MR1360720
  39. W. M. Kantor, ‘On unitary polarities of finite projective planes’, Canad. J. Math. 23 (1971), 1060–1077. MR293491
  40. W. M. Kantor, ‘Homogeneous designs and geometric lattices’, J. Combin. Theory Ser. A 38 (1985), 66–74. MR773556
  41. G. Karpilovsky, The Schur Multiplier (Clarendon Press, Oxford, 1987). MR1200015
  42. P. B. Kleidman, ‘The maximal subgroups of the chevalley groups G_{2}(q) with q odd, the Ree groups ^{2}G_{2}(q), and their automorphism groups’, J. Algebra 117 (1988), 30–71. MR955589
  43. P. B. Kleidman and M. Liebeck, The subgroup structure of the finite classical groups (Cambridge University Press, Cambridge, 1990). MR1057341
  44. G. Korchmáros, ‘Collineation groups doubly transitive on the points at infinity in an affine plane of order 2^{r}’, Arch. Math. (Basel) 37 (1981), 572–576. MR646518
  45. X. Li, ‘A characterization of the finite simple groups’, J. Algebra 245 (2001), 620–649. MR1863895
  46. M. W. Liebeck, ‘On the order of maximal subgroups of the finite classical groups’, Proc. London Math. Soc. (3) 50 (1985), 426–446. MR779398
  47. I. Matulić-Bedenić, ‘The classification of projective planes of order 11 which possess an involution’, Rad. Mat. 1 (1985), 149–157. MR791752
  48. I. Matulić-Bedenić, ‘The classification of projective planes of order 13 which possess an involution’, Rad Hrvatske Akad. Znam. Umjet. 456 (1991), 9–13. MR1211596
  49. H. H. Mitchell, ‘Determination of ordinary and modular ternary linear groups’, Trans. Amer. Math. Soc. 12 (1911), 207–242. MR1500887
  50. B. Mwene, ‘On the subgroups of the group \operatorname{PSL}_{4}(2^{m}) ’, J. Algebra 41 (1976), 79–107. MR409654
  51. D. S. Passman, Permutation Groups (W. A. Benjamin, Inc., New York -Amsterdam, 1968). MR237627
  52. T. Penttila, G. F. Royle and M. K. Simpson, ‘Hyperovals in the known projective planes of order 16’, J. Combin. Des. 4 (1996), 59–65. MR1364099
  53. A. Reifart, ‘The classification of the translation planes of order 16. II’, Geom. Dedicata 17 (1984), 1–9. MR771177
  54. P. Ribenboim, Catalan's conjecture (Acad. Press, Boston, 1994). MR1259738
  55. P. Ribenboim, Fermat's last theorem for amateurs (Springer-Verlag, New York, 1999). MR1719329
  56. R. H. Schulz, ‘Über translationsebenen mit kollineationsgruppen, die die punkte der ausgezeichneten geraden zweinfach transitiv permutieren’, Math. Z. 122 (1971), 246–266. MR287429
  57. R. Shull, ‘Collineations of projective planes of order 9’, J. Combin. Theory Ser. A 37 (1984), 99–120. MR757609
  58. R. Shull, ‘The classification of projective planes of order 9 possessing a collineation group of order 5’, Algebras Groups Geom. 2 (1985), 365–379. MR840903
  59. M. Suzuki, ‘On a class of doubly transitive groups’, Ann. of Math. (2) 75 (1962), 105–145. MR136646
  60. L. Yu and M. Le, ‘On the diophantine equation (x^{n}-1)/(x-1)=y^{m}’, Acta Arith. 21 (1972), 299–301. MR302559