A class of translation planes admitting elations which are not translations

Abstract

We are interested in the extent to which finite t ranslat ion planes can be characterized by the central collineations, other than those with axis 1r162 which they admit . While the class of known translat ion planes is ve ry large, there still is a shortage of kinds of examples. Lff~C~BU~G [2] has shown t h a t in the case of the planes associated with the Suzuki groups, each affine line is the axis of a non-trivial elation. There have been no known examples of t ranslat ion planes of odd order in which more than one point at infinity is on the axis of a non-trivial elation which is no t a translation. The construct ion we give here gives examples of planes with this property. We have shown in [5] tha t , for characteristic greater t han 3, two elations of a t ranslat ion plane which are not translations and have different centers mus t generate a group isomorphic to S L (2, u) for some u. I n our case, the eollineation group has a subgroup isomorphic to GL (2, u) and contains homologies as well as translations. Essentially the same construct ion gives examples of planes which are no t semifield planes in which a single point at infinity is the center of elations which are not translations.