Elations and homologies in collineation groups of finite projective planes

Let G be a collineation group of a finite projective plane. The action of G on the centers and axes of non-identity elations and homologies is discussed. There are several results on the possible numbers of orbits of centers, axes, and center-axis pairs of homologies and elations of a particular order. Several results on the generation of homologies or elations by other homologies or elations reveal additional information on the structures formed by the centers and axes. Some sets of sufficient conditions for the centers and axes to form Desarguesian subplanes are given.