On finite point transitive affine planes with two orbits on \(\ell_{\infty}\) (Q917898)

The purpose of the paper is to study the following conjecture of Kallaher: Let \(\pi\) be a finite affine plane of order n with a collineation group G which is transitive on the affine points of \(\pi\). If G has two orbits on the line at infinity then one of the following statements holds: (I) The plane \(\pi\) is a transition plane and the group G contains the group of translations of \(\pi\), (II) The plane \(\pi\) is a dual translation plane, and the group G contains the group of dual translations of \(\pi\). It contains two theorems, first of them gives some sufficient conditions for a finite affine plane to be a translation plane, and the second theorem describes affine planes of order n with a collineation group which is transitive on affine points and which has two orbits of length 2 and n-1 on the line at infinity.