The derivation of dual translation planes
Let \(\Pi\) be a finite affine plane of order n with a derivable net R, and let \(\Pi^*\) be the plane obtained from \(\Pi\) by deriving R. The group of collineations of \(\Pi\) which map the derivable net R onto itself induces a collineation group on \(\Pi^*\) called the inherited group of \(\Pi^*\). Question: Is the full collineation group of \(\Pi^*\) the inherited group? In general the answer is no; for example, the Hall translation planes are derivable and their derived planes are the desarguesian planes. Consider this question under the additional conditions: (1) the plane \(\Pi\) is a dual translation plane of order \(q^ 2>16\), and (2) the derivable net contains a shears axis. For the known planes satisfying these conditions the answer is yes to the above question. However, if \(\Pi\) is the semifield plane of order 16 with kernel GF(2) and \(\Pi^*\) is the Dempwolff plane of order 16, then the answer is no. The author proves that under the added conditions either the answer is yes or \(\Pi\) and \(\Pi^*\) satisfy certain properties including: (a) q is even, and (b) \(\Pi\) has translations which are not dual translations. (For the full list, see Theorem A of the article. These conditions are satisfied by the Dempwolff plane, but whether there exists a plane of order \(q^ 2>16\) satisfying them is not known.) If \(\Pi_ 1\) and \(\Pi_ 2\) are two dual translation planes with derivable nets \(R_ 1\) and \(R_ 2\) respectively both satisfying conditions (1) and (2), then the article's second major result is that \(\Pi_ 1\) not isomorphic to \(\Pi_ 2\) implies the corresponding derived planes are not isomorphic. The second major thrust of this article is the investigation of the number of non-isomorphic derived planes obtained from the same plane \(\Pi\). The main result in this direction handles the case where the derivable net R is a kernel net. To define kernel net, let \(\Pi\) be a dual translation plane of order \(q^ 2\), and let \(\Pi_ 0\) be the corresponding translation plane; that is, the plane \(\Pi\) is obtained from the (affine) translation plane \(\Pi_ 0\) by deriving with respect to the line \(\ell_{\infty}\) at infinity of \(\Pi_ 0\) and a point V on \(\ell_{\infty}\). If K is a collineation group of order q-1 on \(\Pi\) induced by a group of kernel homologies of \(\Pi_ 0\) and if R is a net in \(\Pi\) left invariant by K, then R is called a kernel net. The author proves: Let \(\Pi_ 1\) and \(\Pi_ 2\) be dual translation planes of the same translation plane \(\Pi_ 0\) with respect to the line \(\ell_{\infty}\) and distinct points \(V_ 1\) and \(V_ 2\) on \(\ell_{\infty}\), let \(\Pi_ 0\) have kernel GF(q), and let \(\Pi^*_ 1\) and \(\Pi^*_ 2\) be obtained from \(\Pi_ 1\) and \(\Pi_ 2\) by deriving with respect to kernel nets \(R_ 1\) and \(R_ 2\) respectively. If the translation complement of \(\Pi_ 0\) consists only of kernel homologies, then \(\Pi^*_ 1\) is not isomorphic to \(\Pi^*_ 2\). It follows that in this case there are \((q^ 2+1)(q+1)\) non-isomorphic planes \(\Pi^*\) obtained by deriving a kernel net in a plane \(\Pi\) dual to \(\Pi_ 0\). Such planes \(\Pi^*\) must be strict semi-translation planes. As one example the author applies this result to the Charnes translation plane of order \(17^ 2\). A more general result given by the author is applied to likeable translation planes; he shows that for a given likeable plane two non-isomorphic strict semi-translation planes can be obtained. The above gives only a flavor of the results and does not do justice to them. More general results are derived, and a beginning has been made on discovering reasonable bounds on the number of non-isomorphic semi- translation planes.
scientific article; zbMATH DE number 4138656
scientific article