A Note on Tangentially Transitive Affine Planes

Let $1 be a finite affine translation plane. If 21 contains a subplane, collineation group pair (2I0, A) such that: * A leaves each point of 2t0 fixed and acts transitively on the points of / \ { / n 2I0}, for / a line of 21 with / n 2l0 a line of 2l0, then 21 is said to be tangentially transitive with respect to 2l0. Jha in his thesis [3] has proved that if 21 is tangentially transitive with respect to 2I0, then 2t has square order p 2m and, with the possible exception of p = 16, 2I0 has order p. Furthermore if 2I0 has order p , then 21 is a generalised Hall plane. In this note we show that up to isomorphism there is exactly one translation plane of order 16, tangentially transitive with respect to a subplane of order 2.-This completes the exceptional case in Jha's theorem. This translation plane has also been constructed by Lorimer [5], but its existence is implicit in the work of Johnson [4], since it is also tangentially transitive with respect to subplanes of order 4 (cf. 3.1) and as such is a generalised Hall plane. I should like to mention that the main result of this note has also been independently proved by N. L. Johnson and T. G. Ostrom.