Collineations of projective planes with different numbers of fixed points and fixed lines (Q1113079)

Let \(F_ P\) and \(F_ L\) be the sets of fixed points and fixed lines of a collineation \(\kappa\) of a projective plane. For the cardinalities \(| F_ P|\) and \(| F_ L|\) it is known that \(| F_ P| =| F_ L|\) if the projective plane is pappian and \(\kappa\) is a projective collineation. The author proves that for every collineation of finite order in a desarguesian projective plane the equality \(| F_ P| =| F_ L|\) holds. There are also shown five examples in desarguesian projective planes with \(| F_ P| <| F_ L|\), having \(F_ P=\emptyset\), \(| F_ L| =l\), or \(| F_ P| =l\), \(F_ L\) the set of the lines through the fixed point, or \(| F_ P| \geq 2\), \(F_ P\) being contained in a line and \(F_ L\) a set of lines through a point of \(F_ P\). One of the examples shows that the equality \(| F_ P| =| F_ L|\) valid for a projective collineation in a pappian plane does not hold in an arbitrary desarguesian projective plane.