Projektivitätengruppen von Translationsebenen (Q788990)

The author succeeds in determining the affine group of projectivities \(\Pi^{aff}\), and the group of projectivities \(\Pi\) in the projective closure, for various classes of finite translation planes \({\mathfrak A}\) of order \(q^ n\). The restriction to translation planes is motivated by the well known fact that the affine group of projectivities of an affine plane consists of all those products of parallel projections which map a fixed line on itself, and that this permutation group contains a regular normal subgroup iff the plane is a translation plane. First, the author shows that \(\Pi^{aff}\) of \({\mathfrak A}\) cannot normalize a field greater than ker \({\mathfrak A}\), the kernel of the quasifield which coordinatizes \({\mathfrak A}\). Then the groups \(\Pi^{aff}\) are determined for planes \({\mathfrak A}\) where \(\Pi^{aff}\) contains a Singer cycle, that is, a cyclic subgroup of \(GL(n,q)\) generated by a cycle of length \(q^ n-1\). If \({\mathfrak A}\) is a finite André plane or a nearfield plane then \(\Pi^{aff}\) consists of all affinities of the vector space belonging to \({\mathfrak A}\), and \(\Pi\) is then an alternating or symmetric group. The same is then proved to hold in some classes of generalized Andrè planes, and also whenever \(n=2\) and \(n=3\) if \(\text{ker} {\mathfrak A}=GF(q)\). If \(\Pi^{aff}\) is solvable then \({\mathfrak A}\) is pappian or the nearfield plane of order 9. The author conjectures: If, for any \({\mathfrak A}\), ker \({\mathfrak A}=GF(q)\), then \(\Pi^{aff}\) is \(AGL(n,q)\); for even \(q\), \(\Pi\) is the alternating group of degree \(q^ n+1\), for odd q it is the symmetric group of degree \(q^ n+1\).