ABSTRACT
Now the question is how do we use the idea of a transversal to a derivable net to create some theory of interest? One way might be to assume that there is a derivable a¢ ne plane that admits a transitivity condition on the components that do not belong to the derivable net. We choose here the concept of a ‘partially ‡ag-transitive a¢ ne plane.’ Recalling that a ‘‡ag’is an incident point-line pair, we consider the ‡ags that are on lines that do not belong to the derivable net and insist that there is a collineation group of the a¢ ne plane that is transitive on these ‡ags. We have shown that derivable nets that admit a transversal can be
embedded either in dual translation planes or translation planes. However, a derivable a¢ ne plane may not be a dual translation plane or a translation plane but still we may use the extension theory to determine properties about the derivable a¢ ne plane, such as orders of groups and general group actions. This means, for example, that incompatibility results on Baer p-groups and elations in order q2 translation planes might be able to be applied to p-groups that act on derivable a¢ ne planes.
