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The Geometry of Elation Groups of a Finite Projective Space

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Abstract

We study the geometry of point-orbits of elation groups with a given center and axis of a finite projective space. We show that there exists a 1-1 correspondence from conjugacy classes of such groups and orbits on projective subspaces (of a suitable dimension) of Singer groups of projective spaces. Together with a recent result of Drudge [7] we establish the number of these elation groups.

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References

  1. Apostol T. M.: Introduction to analytic number theory. Spriger-Verlag, New York (1976)

    MATH  Google Scholar 

  2. Ball S., Blokhuis A., Mazzocca F.: Maximal arcs in Desarguesian planes of odd order do not exist. Combinatorica 17, 31–41 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Buekenhout F.: Existence of unitals in finite translation planes of order q 2 with a kernel of order q. Geometriae Dedicata 5, 189–194 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  4. Casse R., Quinn C. T.: Concerning a characterisation of Buekenhout-Metz unitals. J. Geom. 52, 159–167 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  5. Casse L. R .A., O’Keefe C. M., Penttila T.: Characterizations of Buekenhout-Metz unitals. Geom. Dedicata 59, 29–42 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. Denniston R. H. F.: Some maximal arcs in finite projective planes. J. Combinatorial Theory 6, 317–319 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  7. K. Drudge, On the orbits of Singer groups and their subgroups, Electron. J. Combin. 9 (2002), 10 pp. (electronic).

  8. Ebert G.: The completion problem for partial packings. Geom. Dedicata 18, 261–267 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ebert G., Metsch K., Szönyi T.: Caps embedded in Grassmannians. Geom. Dedicata 70, 181–196 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  10. Glynn D. G.: On a set of lines of PG(3, q) corresponding to a maximal cap contained in the Klein quadric of PG(5, q). Geom. Dedicata 26, 273–280 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hamilton N., Penttila T.: Groups of maximal arcs. J. Combin. Theory Ser. A 94, 63–86 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hirschfeld J. W. P.: Projective geometries over finite fields. Oxford University Press, New York (1998)

    MATH  Google Scholar 

  13. Huppert B.: Endliche Gruppen I. Spriger, Berlin (1967)

    Book  MATH  Google Scholar 

  14. Mathon R.: New maximal arcs in Desarguesian planes. J. Combin. Theory Ser. A 97, 353–368 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Mazzocca F., Polverino O.: Blocking sets in PG(2, q n) from cones of PG(2n, q). J. Algebraic Combin. 2461–81 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Metz R.: On a class of unitals, Geom. Dedicata 8, 125–126 (1979)

    MathSciNet  MATH  Google Scholar 

  17. Polverino O.: Linear sets in finite projective spaces. Discrete Math. 310, 3096–3107 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Singer J.: A theorem in finite projective geometry and some applications to number theory. Trans. Amer. Math. Soc. 43, 377–385 (1938)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica ed Applicazioni, Università degli Studi di Napoli “Federico II”, Via Cintia snc, IT-80126, Napoli, Italy

    Nicola Durante

  2. Dipartimento di Matematica ed Informatica, Università degli Studi della Basilicata, Via dell’Ateneo Lucano, 10, IT-85100, Potenza, Italy

    Alessandro Siciliano

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  1. Nicola Durante
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  2. Alessandro Siciliano
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Correspondence to Alessandro Siciliano.

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Durante, N., Siciliano, A. The Geometry of Elation Groups of a Finite Projective Space. Mediterr. J. Math. 10, 439–448 (2013). https://doi.org/10.1007/s00009-011-0173-1

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  • DOI: https://doi.org/10.1007/s00009-011-0173-1

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