On collineations and dualities of finite generalized polygons (Q624185)

A result of \textit{C. T. Benson} [J. Algebra 15, 443--454 (1970; Zbl 0212.52304)] is generalized to all finite generalized polygons. Given a collineation \(\theta\) of a finite generalized polygon \(S\), a relation between the parameters of \(S\) and, for various natural numbers \(i\), the number of points \(x\) which are mapped to a point at distance \(i\) from \(x\) by \(\theta\) is derived. As a special case, generalized \(2n\)-gons of order \((1,t)\) are also considered and, in the generic case, the exact number of absolute points of a given duality of the underlying generalized \(n\)-gon of order \(t\) is determined.