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

doi:10.1285/i15900932v30n1p13

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

N.L. Johnson

Abstract

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.