Nest replaceable translation planes
A longstanding problem in the field of finite translation planes is the determination of a meaningful classification. Although this is far from being solved in general, substantial progress has occurred for finite translation planes of dimension 2. The article under review is in this vein. Further, it also gives a characterization of planes constructed by nest replacement of a (q-1)-nest of reguli in a desarguesian or hall plane. (For the concept of ``t-nest of reguli'' and nest replacement see \textit{R. D. Baker} and \textit{G. L. Ebert}, J. Lond. Math. Soc., II. Ser. 38, No.2, 341-355 (1988; Zbl 0667.51003).) The principal result is as follows. Let \(\Pi\) be a translation plane of order \(q^ 2\) with GF(q) contained in its kernel. Assume \(\Pi\) has a cyclic collineation group G of order \(q^ 2-1\) in its linear translation complement which has an orbit of length q-1 on the line \(\ell_{\infty}\) at infinity. Then the plane is either a (i) desarguesian plane, (ii) a regular nearfield plane, (iii) constructed from a desarguesian plane by nest replacement of a (q-1)-nest of reguli, or (iv) derived from a plane of type (i), (ii), or (iii). Furthermore, if q is even then \(\Pi\) is either desarguesian or hall (derived from a desarguesian plane). This result plus a second (as yet unpublished) similar result on q-nests by the author and V. Jha show the need for a characterization of nest replacement of t-nests of reguli for arbitrary t.
scientific article; zbMATH DE number 4138657
scientific article