ABSTRACT
In 1972, Ganley [385] showed the existence of unitary polarities in certain finite semifield planes of order q2; certain of the Hughes-Kleinfeld semifield planes, certain of the Knuth semifield planes, as well as the Dickson commutative semifield planes. The unitals then turn out to be parabolic in the sense that the line at infinity is tangent to the unitals. More recently, Abatangelo, Korchma´ros, and Larato [9] have studied ‘transitive parabolic unitals in semifield planes’ (by this we mean that there is a collineation group of the associated semifield plane that fixes the parabolic unital and acts transitively on the affine points). If a translation plane of order q2 admits a collineation group G of order q3 that acts transitively on the affine points of a parabolic unital, the nature of G has not been generally determined. However, Abatangelo, Enea, Korchma´ros, and Larato [8] show that when the plane is a semifield plane coordinatized by a commutative Dickson semifield, then G is never Abelian.
