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  • Regulus  (2)
  • Flag-transitive  (1)
  • 1
    Electronic Resource
    Electronic Resource
    Regulus-free Spreads of PG(3,q) (1996)
    Springer
    Designs, codes and cryptography 8 (1996), S. 79-89 
    Details
    ISSN: 1573-7586
    Keywords: Regulus ; spread ; flag-transitive
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract An old conjecture of Bruck and Bose is that every spreadof Σ = PG(3,q) could be obtained by startingwith a regular spread and reversing reguli. Although it was quicklyrealized that this conjecture is false, at least for qeven, there still remains a gap in the spaces for which it isknown that there are spreads which are regulus-free. In severalpapers Denniston, Bruen, and Bruen and Hirschfeld constructedspreads which were regulus–free, but none of these dealtwith the case when q is a prime congruent to onemodulo three. This paper closes that gap by showing that forany odd prime power q, spreads of PG(3,q) yielding nondesarguesian flag-transitive planes are regulus–free.The arguments are interesting in that they are based on elementarylinear algebra and the arithmetic of finite fields.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1018024723184
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Two-dimensional flag-transitive planes revisited (1996)
    Springer
    Geometriae dedicata 63 (1996), S. 1-15 
    Details
    ISSN: 1572-9168
    Keywords: 51A40 ; 51E15 ; Flag-transitive ; translation plane ; two-dimensional ; enumeration
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper shows that the odd order two-dimensional flag-transitive planes constructed by Kantor-Suetake constitute the same family of planes as those constructed by Baker-Ebert. Moreover, for orders satisfying a modest number theoretical assumption this family consists of all possible such planes of that order. In particular, it is shown that the number of isomorphism classes of (non-Desarguesian) two-dimensional flag-transitive affine planes of order q 2 is precisely (q−1)/2 when q is an odd prime and precisely (q−1)/2e when q=p e is an odd prime power with exponent e that is a power of 2. An enumeration is given in other cases that uses the Möbius inversion formula.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00181182
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Regulus-free spreads ofPG (3,q) (1996)
    Springer
    Designs, codes and cryptography 8 (1996), S. 79-89 
    Details
    ISSN: 1573-7586
    Keywords: Regulus ; spread ; flag-transitive
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract An old conjecture of Bruck and Bose is that every spread of Σ =PG(3,q) could be obtained by starting with a regular spread and reversing reguli. Although it was quickly realized that this conjecture is false, at least forq even, there still remains a gap in the spaces for which it is known that there are spreads which are regulus-free. In several papers Denniston, Bruen, and Bruen and Hirschfeld constructed spreads which were regulus-free, but none of these dealt with the case whenq is a prime congruent to one modulo three. This paper closes that gap by showing that for any odd prime powerq, spreads ofPG(3,q) yielding nondesarguesian flag-transitive planes are regulus-free. The arguments are interesting in that they are based on elementary linear algebra and the arithmetic of finite fields.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00130569
    Location Call Number Expected Availability
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    Paper (German National Licenses)
    Fulltext
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