ABSTRACT
We have seen previously that there are very few types of collineation groups that are known to act on finite translation planes. We mentioned the HeringOstrom theorem that describes the groups generated by the set of affine elations in the translation complement. In the non-solvable case, if the order of the translation plane is pn, the possible groups are isomorphic to SL(2, pt), Sz(2
t), for p = 2 and SL(2, 5) for p = 3 and all of these groups can occur. In particular, there are a variety of translation planes of order pn admitting SL(2, pt), where the p-elements are elations. However, it is also possible that a translation plane of order pn could admit a collineation group in the translation complement isomorphic to SL(2, pt), where the p-elements are say Baer (fix a Baer subplane pointwise). There are important classes of translation planes of order q2 that admit a collineation group SL(2, q) in the translation complement. So, it is of fundamental importance to determine the complete set of translation planes that admit such groups.
