On a theorem of R. A. Liebler

Summary

The conjecture that ‘any affine plane A admitting a collineation group which is rank 3 on points is a translation plane’ was stated by D. G. Higman [3]; M. J. Kallaher [4] proved a slightly weaker statement namely that A is either a translation plane or the dual of a translation plane, and R. A. Liebler [5] overcame the possible ambiguity to prove the conjecture. We offer a proof which, though similar to [4] and [5] in the first stages, completes the final steps more smoothly. Thus we prove

Theorem (Liebler). An affine plane A admitting a collineation group H which is rank 3 on points is a translation plane, (and H contains all translations).

Let A be an affine plane of order k, with projective completion P having line at infinity l(∞). P(∞), P(l), …, P(k) are the points of l(∞) and l(l), …, l(k) the affine lines through P(∞). For a point P and line l of P, (P) denotes the set of lines through P and (l) the set of points on l. If G is a group acting on the set Ω, then G^Ω is the natural permutation group induced by G on Ω.