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On some extension of 1-spread sets. (English) Zbl 0646.51006
The author studies a class of translation planes of order q 4 with the properties (1) the linear translation complement LC(\(\pi)\) has a shears group P of order at least q 2 and (2) LC(\(\pi)\) has a Baer subgroup Q of order \(q+1\) with [P,Q]\(\neq 1.\)
A set \(\Sigma\) of q 2 (2,2)-matrices over \(K=GF(q)\) is said to be a 1- spread set if it contains the zero matrix 0 and X-Y is nonsingular for any distinct X,Y\(\in \Sigma\). Let \(F=GF(q\) 2)\(\supset K\). If \(\Sigma\) ’ is an arbitrary 1-spread set over K, then \(\Sigma '=\{\left( \begin{matrix} x\\ g(x,y)\end{matrix} \begin{matrix} y\\ h(x,y)\end{matrix} \right)|\) x,y\(\in K\}\) for suitable maps g,h from \(K\times K\) to K. Let \(\Sigma_ f=\{\left( \begin{matrix} x\\ f(y)\end{matrix} \begin{matrix} y\\ x\quad q\end{matrix} \right)|\) x,y\(\in F\}\) when \(f: F\to F\) is a map such that \(f(x+yt)=g(x,y)-h(x,y)t\) for x,y\(\in K\). Then \(\Sigma_ f\) is a 1-spread set over F and the resulting translation plane of order q 4 with kernel F, say \(\pi\), has the properties (1) and (2). Let \(\Omega\) (F) be the set of maps from F to itself satisfying \(f(0)=0\), \((x-y)(f(x)-f(y))\not\in K\) for any distinct x,y\(\in F\). Let \(\pi\) (F) denote the set of planes \(\pi_ f\) corresponding to \(\Sigma_ f\) with \(f\in \Omega (F)\). Then \(\pi\) (F) is characterized in this paper as the set of translation planes with kernel F having properties (1) and (2). It is also proved that the translation complements of these planes are solvable when \(p>2\).
Reviewer: T.Thrivikraman

51A40 Translation planes and spreads in linear incidence geometry