On translation planes of order \(q^ 2\) which admit an autotopism group having an orbit of length \(q^ 2-q\) (Q1066468)

In a translation plane \(\pi\), define a linear autotopism group as a subgroup of the linear translation complement of \(\pi\) which fixes at least two points on the line at infinity. The paper deals with translation planes of order \(q^ 2\) having certain properties. First, the quasifields corresponding to these planes are studied. Let K be a field. Let h(x) map K into K, and let r(y) and s(y) map \(K\setminus \{0\}\) into K. Set \(f(x,y)=-y^{-1}(x^ 2-r(y)x-s(y))\) and \(g(x,y)=-x+r(y).\) After imposing certain conditions on r, s, and h, \({\tilde \Phi}{}_ K\) is defined as the set of triples (r,s,h), and \(\Phi_ K:=\{(r,s,h)| \quad (r,s,h)\in {\tilde \Phi}_ K,\quad h(x)=0\quad for\quad any\quad x\in K\}.\) Then a multiplication is defined in a quasifield \(Q_{(r,s,h)}\), which is a 2-dimensional left vector space over K. For finite K the following theorem is obtained: If \(\pi\) has order \(q^ 2\) and has GF(q) in its kernel, then it is coordinatized by a \(Q_{(r,s,h)}\) with (r,s,h)\(\in {\tilde \Phi}_{GF(q)}\) if and only if \(\pi\) admits a linear autotopism group of order q. Another theorem, with the same hypotheses, states: If \(\pi\) admits a linear autotopism group having an orbit of length \(q^ 2\)-q on the line of infinity, then \(\pi\) is coordinatized by a \(Q_{(r,s,0)}\) with \((r,s,0)\in \Phi_ K\), where \(r(y)=ay^ n\) and \(s(y)=by^{2n}\) for suitable a and b in K and integral n, \(0\leq n<q-1.\) Finally, for the planes mentioned in the last theorem, the linear complement is determined for \(n\not\in \{e(q-1)|\) \(e=0,1/2,1/3,2/3,1/4,3/4\}.\)