Regular parallelisms from translation planes

Abstract

Let π be a translation plane of order q 4 admitting a collineation group G isomorphic to SL(2, q), p r = q, in the translation complement. It is shown that if the p-elements of G are elations and L is an elation axis then the kernel of π is GF( q) and G acts 1 2 - transitive on l ∞− N∩l ∞ (where N denotes the elation net) if and only if there is a regular parallelism of L (considering L as PG(3, q)). With no assumption on the p-elements, it is shown that the existence SL(2, q)× z 1+ q+ q 2 as a collineation group implies that the p-elements are elations, the kernel is GF( q) and any elation axis L admits a regular parallelism as PG(3, q).