Translation planes of odd order and odd dimension

The author considers one of the main problems in finite translation planes to be the identification of the abstract groups which can act as collineation groups and how those groups can act. The paper is concerned with the case where the plane is defined on a vector space of dimension 2 d over G F ( q ) , where q and d are odd. If the stabilizer of the zero vector is non-solvable, let G 0 be a minimal normal non-solvable subgroup. We suspect that G 0 must be isomorphic to some S L ( 2 , u ) or homomorphic to A 6 or A 7 . Our main result is that this is the case when d is the product of distinct primes. The results depend heavily on the Gorenstein-Walter determination of finite groups having dihedral Sylow 2 -groups when d and q are both odd. The methods and results overlap those in a joint paper by Kallaher and the author which is to appear in Geometriae Dedicata. The only known example (besides Desarguesian planes) is Hering's plane of order 27 (i.e., d and q are both equal to 3 ) which admits S L ( 2 , 13 ) .