Characterization of translation planes by orbit lengths
\textit{M. Walker} [Abh. Math. Semin. Univ. Hamb. 49, 216-233 (1979; Zbl 0444.51005)] showed that a translation plane of odd order \(q^2\) with spread in \(\text{PG} (3,q)\) that admits a collineation group in the translation complement isomorphic to \(\text{SL} (2,q)\) is Desarguesian, a Hall plane, a Hering plane, or one of the three exceptional Walker planes of order 25. If \(q\neq 5\), then \(\text{SL} (2,q)\) operates on the line at infinity with one orbit of length \(q+1\) and \(i\) orbits of length \((q^2-q)/i\) where \(i=1\) or 2. In the paper under review the authors show that this orbit configuration determines the translation plane. They consider a translation plane \(\pi\) of odd order \(q^2\) with spread in \(\text{PG} (3,q)\) and assume 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 \(i=1,2\). The authors then show that \(\pi\) is (i) Desarguesian, \(G\) modulo the kernel is reducible and there exists an elation, and \(i=1\); (ii) a Hall plane, \(G\) is reducible and there exists a Baer \(p\)-element, \(q=p^r\), and \(i=1\); (iii) a Hering plane, \(G\) is irreducible, \(q=p^r\) for \(r\) odd, and \(i=2\) (\(i=1\) or 2 when the order is 25); (iv) a derived likeable Walker plane of order 25. A careful analysis and repeated use of the classification of finite affine translation planes of order \(n\) admitting a collineation group that has a point orbit of length \(n\) and induces upon it a non-solvable doubly transitive group by \textit{M. Ganley, V. Jha} and \textit{N. L. Johnson} [The translation planes admitting a non-solvable doubly transitive line-sized orbit, to appear in J. Geom.] show that the existence of an elation or a Baer \(p\)-element leads to a Desarguesian or Hall plane. If all \(p\)-elements are quartic, one obtains a Hering plane or Walker plane of order 25. If one of the exceptional orders \(3^4\), \(3^6\), \(11^2\), \(19^2\), \(29^2\) or \(59^2\) in the classification by Ganley, Jha, Johnson occurs, then again the plane is Desarguesian or a Hall plane.
scientific article; zbMATH DE number 1383256