Dual parallelisms

Abstract

Assume that $\rho$ is a parallelism in $PG(3,K)$, for $K$ a field, that admits a collineation group $G$ that fixes one spread $\Sigma$ and acts transitively on the remaining spreads of $\rho$. If $G$ contains suitable central collineations of $\Sigma$ then it is shown that the dual parallelism is a parallelism that can never be isomorphic to the original. The results show that the Johnson parallelisms of Hall or Knuth type, the Johnson-Pomareda parallelisms of type $f$ and all of the "derived" parallelisms produce dual parallelisms which are parallelisms but are nonisomorphic to the original parallelism.