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.
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Communicated by: D. Jungnickel
Dedicated to Hanfried Lenz on the occasion of his 80th birthday
This work was partially supported by NSA grant MDA 904-95-H-1013.
This work was partially supported by NSA grant MDA 904-94-H-2033.
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Baker, R.D., Ebert, G.L. Regulus-free spreads ofPG (3,q). Des Codes Crypt 8, 79–89 (1996). https://doi.org/10.1007/BF00130569
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DOI: https://doi.org/10.1007/BF00130569