ABSTRACT
The irregular nearfield planes of order p2, admit affine homology groups isomorphic to either SL(2, 5) or SL(2, 3) in their corresponding sharply transitive groups. Furthermore, by taking the product G of the two affine homology groups with symmetric axes and coaxes, we obtain a collineation group with the property that given any non-axis (of an affine homology) N , then GN is transitive on the non-zero vectors of N . We note that in the irregular nearfield planes there is a collineation of the form (x, y) 7−→ (y, x) if the axes are called x = 0, y = 0. In particular, this means that the irregular nearfield planes are of rank 3. Indeed, if a rank 3 translation plane has an orbit of length 2 on the line at infinity (i.e., interchanges say x = 0, y = 0), usually the translation plane is a generalized Andre´ and only is not when possibly the order is in {5, 7, 11, 23, 29, 59} . Since the irregular nearfield planes are not generalized Andre´ planes, we see the restriction on orders is strict. But, in particular, rank 3 translation planes with an orbit of length 2 are interesting and actually may be classified. The classification involves the analysis of some exceptional planes. The work of Kallaher, Ostrom [844] and Lu¨neburg [960] (together with the computer analysis of all translation planes of order 72) gives the following result:
Theorem 9.1. (Kallaher-Ostrom [844], Lu¨neburg [960] + translation planes of order 72). Let pi be a rank 3 translation plane with an orbit of length 2 on the line at infinity. Then either the plane is a generalized Andre´ plane, one of the irregular nearfield planes, or the exceptional Lu¨neburg plane of order 72 (see below for the description).
