Characterization of translation planes by orbit lengths. II: Even order
In [Geom. Dedicata 78, 69-80 (1999; Zbl 0941.51006)] the authors considered translation planes \(\pi\) of odd order \(q^2\) with spread in \(\text{PG}(3,q)\) and assumed that \(\pi\) admits a linear collineation group \(G\) such that on the line at infinity \(G\) has one point orbit of length \(q+1\) and \(i\) orbits of length \((q^2-q)/i\) for \(i=1,2\). They showed that \(\pi\) is Desarguesian, a Hall plane, a Hering plane, or a derived likeable Walker plane of order 25. In the paper under review the authors continue their investigations and consider the even order case. They show that the Desarguesian and Hall planes are the only translation planes of even order \(q^2\) that admit a collineation group \(G\) in the translation complement such that on the line at infinity \(G\) has point orbits of length \(q+1\) and \(q^2-q\). As an application of this result they further show that a translation plane of even order \(q^2\) with spread in \(\text{PG}(3,q)\) that admits a linear collineation group \(G\) such that on the line at infinity \(G\) has one point orbit of length \(q+1\) and \(i\) orbits of length \((q^2-q)/i\) for \(i=1,2\) is Desarguesian, a Hall plane, or an Ott-Schaeffer plane. In the former two kinds of planes one has \(i=1\) and in the latter kind of plane one has \(i=2\). The classification of translation planes of order 16 by \textit{U. Dempwolff} and \textit{A. Reifart} [Geom. Dedicata 15, 137-153 (1983; Zbl 0532.51003)] implies both results in case of order 16. To prove the first result order 64 requires special considerations. In the general case, central involutions of a Sylow 2-subgroup of \(G\) and their possible fixed-point configurations are studied to eventually show that the translation plane admits \(SL(2,q)\) as a collineation group. Such planes were classified by \textit{D. A. Foulser} and the last author [J. Algebra 86, 385-406 (1984; Zbl 0527.51012)] and must be Desarguesian, Hall, Ott-Schaeffer, or the Dempwollff plane of order 16. The assumptions on the orbit lengths then rule out the latter two planes.
scientific article; zbMATH DE number 1547312