On a class of translation planes of square order

Abstract

A class of translation planes of order q 2, where q = p r , p is a prime, p ⩾7, p ≠± 1 (mod 10) and r is an odd natural number is constructed and the translation complements of these planes are determined. A property shared by all these planes is that the translation complement fixes a distinguished point and divides the remaining distinguished points into two orbits of length q and q 2 − q. The order of the translation complement is rq( q − 1) 2 except for q = 7 and q = 13. The translation complements of these exceptional cases are also briefly studied. The class of planes considered in this paper are distinct from the classes of translation planes of S.D. Cohen and M.J. Ganley [Quart. J. Math. Oxford, 35 (1984) 101–113].