The translation planes admitting a nonsolvable doubly transitive line-sized orbit
One of the oldest type of problems in the theory of finite (affine) translation planes is: Characterize all such planes having a collineation group with a given property. For example, \textit{D. A. Foulser} and \textit{N. L. Johnson} in two articles [J. Algebra 86, 385-406 (1984; Zbl 0527.51012) and J. Geom. 18, 122-139 (1982; Zbl 0527.51013)] characterize translation planes of order \(q^2\) admitting \(\text{SL} (2,q)\) as a collineation group. In this article the authors consider a translation plane \(\Pi\) of order \(n\) admitting a nonsolvable collineation group \(G\) fixing a set \(\Omega\) of \(n\) points in \(\Pi\) and acting doubly transitive on \(\Omega\). If \(n \neq 3^4,3^6, 11^2,19^2, 29^2,59^2\) then the authors prove \(\Pi\) is desarguesian, Hall, the Dempwolff plane of order 16, or one of three Walker planes of order 25. With regard to the excluded orders note that for the last four orders there are the exceptional nearfield planes and the Lüneburg planes of type \(Rp\). [See \textit{H. Lüneburg}, `Translation planes', Springer-Verlag (1980; Zbl 0446.51003), Chapter III.] For \(n=3^6\) the only known possibilities are the desarguesian and Hall planes; for \(n=3^4\), besides the desarguesian and Hall planes there are the Prohaska plane and dual Prohaska, plane. [See \textit{O. Prohaska}, Arch. Math. 28, 550-556 (1977; Zbl 0354.50015).] The authors first reduce the problem to a situation where they can apply the classification of \textit{C. Hering} [Geom. Dedicata 2, 425-460 (1974; Zbl 0292.20045)] on transitive linear groups in \(\text{GL}(d,p)\). Then using known results they eliminate most of the groups appearing in Hering's list. In particular, they use results concerning large Baer subgroups and large planar \(p\)-groups. Finally, they are reduced to the case where \(G\cong\text{SL}(2,q)\) and they can apply the Foulser-Johnson theorem.
scientific article; zbMATH DE number 1565896
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