On two-transitive ovals in projective planes of even order (Q1894905)

Let \(\pi\) be a finite projective plane of even order \(n\). Let \(G\) be a collineation group of \(\pi\) fixing an oval \(\Omega\) and acting 2- transitively on its points. The paper deals in some detail with one of the three possibilities obtained for \(G\) by \textit{M. Biliotti} and the second author [Ann. Discrete Math. 30, 85-97 (1986; Zbl 0601.51012)] namely the case where \(G\) also fixes a line \(\ell\) which is external to \(\Omega\). It is proved that either \(n \in \{2,4\}\) or \(n \equiv 0 \mod 8\) and the Sylow 2- subgroups of \(G\) are generalized quaternion groups. The proof makes use of some features of the action of \(G\) on \(\pi\), in particular properties of the involutions and the existence of a \(G\)-invariant family of ovals with common knot (including \(\Omega\)) partitioning the points not on \(\ell\) and distinct from the knot.