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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

MSC:
 51A40 Translation planes and spreads in linear incidence geometry