Transitive projective planes and insoluble groups

Gill, Nick

doi:10.1090/tran/6366

Transitive projective planes and insoluble groups
HTML articles powered by AMS MathViewer

by Nick Gill PDF

Trans. Amer. Math. Soc. 368 (2016), 3017-3057 Request permission

Abstract:

Suppose that a group $G$ acts transitively on the points of $\mathcal {P}$, a finite non-Desarguesian projective plane. We prove that if $G$ is insoluble, then $G/O(G)$ is isomorphic to $\mathrm {SL}_2(5)$ or $\mathrm {SL}_2(5).2$.

References

M. Aschbacher, On the maximal subgroups of the finite classical groups, Invent. Math. 76 (1984), no. 3, 469–514. MR 746539, DOI 10.1007/BF01388470
M. Aschbacher, Finite group theory, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 10, Cambridge University Press, Cambridge, 2000. MR 1777008, DOI 10.1017/CBO9781139175319
N. Burgoyne, R. Griess, and R. Lyons, Maximal subgroups and automorphisms of Chevalley groups, Pacific J. Math. 71 (1977), no. 2, 365–403. MR 444795
Michael J. Collins, Modular analogues of Jordan’s theorem for finite linear groups, J. Reine Angew. Math. 624 (2008), 143–171. MR 2456628, DOI 10.1515/CRELLE.2008.084
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, $\Bbb {ATLAS}$ of finite groups, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups; With computational assistance from J. G. Thackray. MR 827219
Peter Dembowski, Finite geometries, Classics in Mathematics, Springer-Verlag, Berlin, 1997. Reprint of the 1968 original. MR 1434062
Leonard Eugene Dickson, Linear groups: With an exposition of the Galois field theory, Dover Publications, Inc., New York, 1958. With an introduction by W. Magnus. MR 0104735
Nick Gill, Nilpotent Singer groups, Electron. J. Combin. 13 (2006), no. 1, Research Paper 94, 6. MR 2274309
Nick Gill, Transitive projective planes, Adv. Geom. 7 (2007), no. 4, 475–528. MR 2360899, DOI 10.1515/ADVGEOM.2007.030
George Glauberman, Central elements in core-free groups, J. Algebra 4 (1966), 403–420. MR 202822, DOI 10.1016/0021-8693(66)90030-5
Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, Mathematical Surveys and Monographs, vol. 40, American Mathematical Society, Providence, RI, 1994. MR 1303592, DOI 10.1090/surv/040.1
Daniel Gorenstein, Finite groups, Harper & Row, Publishers, New York-London, 1968. MR 0231903
Daniel M. Gordon, The prime power conjecture is true for $n<2,000,000$, Electron. J. Combin. 1 (1994), Research Paper 6, approx. 7. MR 1293396
I. Martin Isaacs, Finite group theory, Graduate Studies in Mathematics, vol. 92, American Mathematical Society, Providence, RI, 2008. MR 2426855, DOI 10.1090/gsm/092
William M. Kantor, Primitive permutation groups of odd degree, and an application to finite projective planes, J. Algebra 106 (1987), no. 1, 15–45. MR 878466, DOI 10.1016/0021-8693(87)90019-6
Peter Kleidman and Martin Liebeck, The subgroup structure of the finite classical groups, London Mathematical Society Lecture Note Series, vol. 129, Cambridge University Press, Cambridge, 1990. MR 1057341, DOI 10.1017/CBO9780511629235
Martin W. Liebeck, On the orders of maximal subgroups of the finite classical groups, Proc. London Math. Soc. (3) 50 (1985), no. 3, 426–446. MR 779398, DOI 10.1112/plms/s3-50.3.426
Attila Maróti, Bounding the number of conjugacy classes of a permutation group, J. Group Theory 8 (2005), no. 3, 273–289. MR 2137971, DOI 10.1515/jgth.2005.8.3.273
T. G. Ostrom and A. Wagner, On projective and affine planes with transitive collineation groups. , Math. Z 71 (1959), 186–199. MR 0110975, DOI 10.1007/BF01181398
P. P. Pálfy, A polynomial bound for the orders of primitive solvable groups, J. Algebra 77 (1982), no. 1, 127–137. MR 665168, DOI 10.1016/0021-8693(82)90281-2
Donald Passman, Permutation groups, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0237627
Cheryl E. Praeger and Jan Saxl, On the orders of primitive permutation groups, Bull. London Math. Soc. 12 (1980), no. 4, 303–307. MR 576980, DOI 10.1112/blms/12.4.303
Koen Thas, Finite flag-transitive projective planes: a survey and some remarks, Discrete Math. 266 (2003), no. 1-3, 417–429. The 18th British Combinatorial Conference (Brighton, 2001). MR 1991732, DOI 10.1016/S0012-365X(02)00823-3
Koen Thas and Don Zagier, Finite projective planes, Fermat curves, and Gaussian periods, J. Eur. Math. Soc. (JEMS) 10 (2008), no. 1, 173–190. MR 2349900, DOI 10.4171/JEMS/107
A. Wagner, On perspectivities of finite projective planes, Math. Z 71 (1959), 113–123. MR 0110974, DOI 10.1007/BF01181392
Daniel Gorenstein and John H. Walter, The characterization of finite groups with dihedral Sylow $2$-subgroups. I, J. Algebra 2 (1965), 85–151. MR 177032, DOI 10.1016/0021-8693(65)90027-X
Hans Zassenhaus, Über endliche Fastkörper, Abh. Math. Sem. Univ. Hamburg 11 (1935), no. 1, 187–220 (German). MR 3069653, DOI 10.1007/BF02940723