COLLINEATION GROUPS OF TRANSLATION PLANES OF SMALL DIMENSION

ABSTRACT. A subgroup of the linear translation complement of a translation plane is geometrically irreducible if it has no invariant lines or subplanes. A similar defi-nition can be given for "geometrically primitive". If a group is geometrically pri-mitive and solvable then it is fixed point free or metacyclic or has a normal subgroup 2a+b a of order w where w divides the dimension of the vector space. Similar conditions hold for solvable normal subgroups of geometrically primitive nonsolvable groups. When the dimension of the vector space is small there are restrictions on the group which might possibly be in the translation complement. We look at the situation for certain orders of the plane. KEY WORDS AND PHRASES. Translon plan, translaon complement, linear groups. 1980 MATHEMATICS SUBJECT CLASSIFICATION CODES. 51A40, 20H30. i. INTRODUCTION. dA translation plane of order q with kernel GF(q) F can be represented as follows: Let V be a vector space of dimension 2d over F. A spread defined on V is