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Simple groups acting two-transitively on the set of generators of a finite elation Laguerre plane

  • Günter F. Steinke
  • Markus J. Stroppel
Original Paper

Abstract

We show that finite elation Laguerre planes with a group of automorphisms acting two-transitively on the set of generators are Miquelian. In this paper we discuss in detail three series of groups of (twisted) Lie type that may possibly occur as socles of the two-transitive group induced on the set of generators; all other cases have been treated in a separate paper.

Keywords

Laguerre plane Elation group Two-transitive group Socle  Simple group of Lie type Unitary group Ree group 

Mathematics Subject Classification (2000)

51E25 51B15 20B20 

Notes

Acknowledgments

The present investigation has been conducted during a stay of the second author as a Visiting Erskine Fellow at the University of Canterbury, Christchurch, New Zealand.

References

  1. Alperin, J.L., Gorenstein, D.: The multiplicators of certain simple groups. Proc. Am. Math. Soc. 17, 515–519 (1966). doi: 10.2307/2035202 CrossRefzbMATHMathSciNetGoogle Scholar
  2. Aschbacher, M.: Finite group theory. Cambridge Studies in Advanced Mathematics, vol 10, 2nd edn. Cambridge University Press, Cambridge (2000)Google Scholar
  3. Baer, R.: Projectivities with fixed points on every line of the plane. Bull. Am. Math. Soc. 52, 273–286 (1946). doi: 10.1090/S0002-9904-1946-08557-2 CrossRefzbMATHMathSciNetGoogle Scholar
  4. Casse, L.R.A., Thas, J.A., Wild, P.R.: \((q^n+1)\)-sets of PG\((3n-1, q)\), generalized quadrangles and Laguerre planes. Simon Stevin 59(1), 21–42 (1985)zbMATHMathSciNetGoogle Scholar
  5. Chen, Y., Kaerlein, G.: Eine Bemerkung über endliche Laguerre- und Minkowski-Ebenen. Geom. Dedic. 2, 193–194 (1973). doi: 10.1007/BF00147856 CrossRefzbMATHMathSciNetGoogle Scholar
  6. Czerwinski, T.: Finite translation planes with collineation groups doubly transitive on the points at infinity. J. Algebra 22, 428–441 (1972). doi: 10.1016/0021-8693(72)90159-7 CrossRefzbMATHMathSciNetGoogle Scholar
  7. Dieudonné, J.A.: La géométrie des groupes classiques, Ergebnisse der Mathematik und ihrer Grenzgebiete (N.F.), vol. 5, 3rd edn. Springer, Berlin (1971)Google Scholar
  8. Griess Jr, R.L.: Schur multipliers of finite simple groups of Lie type. Trans. Am. Math. Soc. 183, 355–421 (1973). doi: 10.2307/1996474 CrossRefzbMATHMathSciNetGoogle Scholar
  9. Grundhöfer, T., Krinn, B., Stroppel, M.J.: Non-existence of isomorphisms between certain unitals. Des. Codes Cryptogr. 60(2), 197–201 (2011). doi: 10.1007/s10623-010-9428-2 CrossRefzbMATHMathSciNetGoogle Scholar
  10. Grundhöfer, T., Stroppel, M.J., Van Maldeghem, H.: Unitals admitting all translations. J. Combin. Des. 21(10), 419–431 (2013). doi: 10.1002/jcd.21329 Google Scholar
  11. Hall Jr, M.: Steiner triple systems with a doubly transitive automorphism group. J. Combin. Theory Ser. A 38(2), 192–202 (1985). doi: 10.1016/0097-3165(85)90069-X CrossRefzbMATHMathSciNetGoogle Scholar
  12. Hiramine, Y.: On finite affine planes with a \(2\)-transitive orbit on \(l_\infty \). J. Algebra 162(2), 392–409 (1993). doi: 10.1006/jabr.1993.1262
  13. Huppert, B.: Endliche Gruppen. I, Die Grundlehren der Mathematischen Wissenschaften, vol 134. Springer, Berlin (1967)Google Scholar
  14. Huppert, B., Blackburn, N.: Finite groups. III, Grundlehren der Mathematischen Wissenschaften, vol 243. Springer, Berlin (1982)Google Scholar
  15. Jacobson, N.: Basic Algebra. II, 2nd edn. W. H. Freeman and Company, New York (1989)Google Scholar
  16. Joswig, M.: Pseudo-ovals, elation Laguerre planes, and translation generalized quadrangles. Beitr. Algebra Geom. 40(1):141–152 (1999). http://www.emis.de/journals/BAG/vol.40/no.1/11.html
  17. Kantor, W.M.: Homogeneous designs and geometric lattices. J. Combin. Theory Ser. A 38(1), 66–74 (1985). doi: 10.1016/0097-3165(85)90022-6 CrossRefzbMATHMathSciNetGoogle Scholar
  18. Lüneburg, H.: Some remarks concerning the Ree groups of type \((G_{2})\). J. Algebra 3, 256–259 (1966). doi:  10.1016/0021-8693(66)90014-7 CrossRefzbMATHMathSciNetGoogle Scholar
  19. Payne, S.E., Thas, J.A.: Finite generalized quadrangles, 2nd edn. EMS Series of Lectures in Mathematics. European Mathematical Society (EMS), Zürich (2009) doi: 10.4171/066
  20. Ree, R.: A family of simple groups associated with the simple Lie algebra of type \((G_{2})\). Am. J. Math. 83, 432–462 (1961). doi:  10.2307/2372888 CrossRefzbMATHMathSciNetGoogle Scholar
  21. Schulz, R.-H.: Über Translationsebenen mit Kollineationsgruppen, die die Punkte der ausgezeichneten Geraden zweifach transitiv permutieren. Math. Z. 122, 246–266 (1971). doi: 10.1007/BF01109919 CrossRefzbMATHMathSciNetGoogle Scholar
  22. Steinke, G.F.: On the structure of finite elation Laguerre planes. J. Geom. 41(1–2), 162–179 (1991). doi: 10.1007/BF01258517 CrossRefzbMATHMathSciNetGoogle Scholar
  23. Steinke, G.F.: A remark on Benz planes of order \(9\). Ars. Combin. 34, 257–267 (1992)zbMATHMathSciNetGoogle Scholar
  24. Steinke, G.F.: Elation Laguerre planes of order \(p^2\) that admit an automorphism group of order \(p^2\) in the elation complement. Australas. J. Combin. 8, 77–98 (1993). http://ajc.maths.uq.edu.au/pdf/8/ajc-v8-p77.pdf
  25. Steinke, G.F., Stroppel, M.J.: Finite elation Laguerre planes admitting a two-transitive group on their set of generators. Innov. Incidence Geom. (2013, to appear)Google Scholar
  26. Thas, J.A.: The \(m\)-dimensional projective space \(S_{m}(M_{n}\)(GF(\(q\)))) over the total matrix algebra \(M_{n}\)(GF(\(q\))) of the \(n\times n\)-matrices with elements in the Galois field GF(\(q\)). Rend. Mat. (6) 4, 459–532 (1971)Google Scholar
  27. Thas, J.A., Thas, K., Van Maldeghem, H.: Translation generalized quadrangles. Series in Pure Mathematics, vol 26. World Scientific Publishing Co. Pte. Ltd., Hackensack (2006). doi: 10.1142/9789812772916
  28. Zsigmondy, K.: Zur Theorie der Potenzreste. Monatsh. Math. Phys. 3(1), 265–284 (1892). doi: 10.1007/BF01692444 CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© The Managing Editors 2013

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of CanterburyChristchurchNew Zealand
  2. 2.LExMath Fakultät für Mathematik und PhysikUniversität StuttgartStuttgartGermany

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