The non-existence of desarguesian t-parallelisms, t an odd prime


In this article, it is shown that when t is a prime, partial Desarguesian %t-parallelisms in PG(zt-1,q) of m t-spreads are equivalent totranslation nets of order q^{zt} and degree 1+q+m(q^{zt}-q). Thus, abound is established for the \ number of t-spreads in partial Desarguesian t-parallelisms, when t is an odd prime. This also shows that therecannot be Desarguesian t-parallelisms when t is an odd prime.

DOI Code: 10.1285/i15900932v30n1p13

t-parallelism; translation planes; rational Pappian partial spreads

Classification: 51E20; 51A40

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