Dual parallelisms

Abstract

Assume that is a parallelism in , for a field, that admits a collineation group that fixes one spread and acts transitively on the remaining spreads of . If contains suitable central collineations of 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 and all of the "derived" parallelisms produce dual parallelisms which are parallelisms but are nonisomorphic to the original parallelism.