Translation planes of order \(q^ 2\) that admit \(q+1\) elations
The main theorem on derivable translation planes is: ''Let \(\pi\) be a translation plane of order \(q^ 2\), 8\(\neq q\neq 9\), and of dimension 2 over GF(q). Let \(\pi\) contain a derivable net \({\mathcal N}\) and for each component \(\ell\) of \({\mathcal N}\) assume there is an elation with axis \(\ell\) leaving \({\mathcal N}\) invariant. Then there is a Desarguesian plane \(\Sigma\) such that \(\pi\) is obtained by multiple derivation in \(\Sigma\) ; and \(\pi\) may be constructed by replacing an odd number of pairwise disjoint derivable sets in \(\Sigma\).'' Let now \({\mathcal N}\) be a net in the translation plane. Let \(\Sigma\) be a Desarguesian plane such that each of the components of \({\mathcal N}\) is a Baer subplane in a derivable net of \(\Sigma\) sharing two fixed components. If \({\mathcal N}\) is not derivable, \({\mathcal N}\) is said to be twisted through \(\Sigma\). The author proves that either the elation net \({\mathcal N}\) is derivable, and then the main theorem applies, or \({\mathcal N}\) is twisted through a Desarguesian plane \(\Sigma\). The paper contains many examples.
scientific article; zbMATH DE number 3855861