On Desarguesian extensions of elation nets
In ''A characterization of the Desarguesian planes of order \(q^ 2\) by SL(2,q)'' [Int. J. Math. Math. Sci. 6, 605-608 (1983; Zbl 0519.51002)] \textit{D. A. Foulser}, \textit{N. L. Johnson} and \textit{T. G. Ostrom} proved: Let \(\pi\) be a translation plane of order \(p^{2r}\), p prime, that admits a collineation group in the translation complement which is isomorphic to \(SL(2,p^ r)\) where the p-elements are elations. Then \(\pi\) is Desarguesian. By adapting the proof of this theorem N. L. Johnson obtained the following more general result: Let \(\pi\) be a translation plane of order \(p^{2rs}\) that admits a collineation group G in the translation complement which is ismorphic to \(SL(2,p^ r)\) where the p-elements are elations. Let \({\mathcal N}\) denote the elation net of degree \(1+p^ r\). Then there is always a rational Desarguesian extension \({\mathcal M}\) of \({\mathcal N}\) (of order \(p^{2r})\) within \(\pi\). Furthermore, (\({\mathcal M}-{\mathcal N})\cap l_{\infty}\) is a G-orbit and, conversely, any orbit of length \(p^{2r}-p^ r\) defines an extension. This theorem has the following corollaries: (1) Let \(\pi\) be a translation plane of order \(q^ 4\) where \(q=p^ r\), p prime. If \(\pi\) admits a collineation group in the translation complement which is isomorphic to SL(2,q) where the p- elements are elations, then \(\pi\) is derivable and the derivable net \({\mathcal D}\) contains the elation net of degree \(q+1\), (2) Let \(\pi\) be any translation plane of order \(p^ 4\), p prime. If \(\pi\) admits elations with at least two nonparallel axes then \(\pi\) is derivable and the derivable net contains at least \(1+p\) elation axes, and (3) Let \(\pi\) be a translation plane of order \(p^ 4\), p prime, that admits elations with at least two components. Then there is a corresponding plane \({\bar\pi }\) that admits SL(2,q) where the p-elements are Baer.
scientific article; zbMATH DE number 3872161
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