Some consequences of a result of Brouwer (Q2715950)

A generalized \(n\)-gon is a point-line geometry the incidence graph of which has girth \(2n\) and diameter \(n\). A generalized \(n\)-gon is said to be thick if every element is incident with at least three other elements. Let \(\Gamma \) be a finite generalized \(n\)-gon. A \(k\)-path is a path on \(k+1\) vertices in the incidence graph of \(\Gamma \). A \((2n-1)\)-path the extremities of which are incident is an ordered apartment. The elements of an ordered apartment form an ordinary apartment. The polygon \(\Gamma \) is a \(k\)-Moufang polygon if, for any \(k\)-path \(\gamma \), the collineation group of \(\Gamma \) fixing every element incident with an inner vertex of \(\gamma \) acts transitively on the set of apartments containing all elements of \(\gamma \). A Moufang polygon is an \(n\)-Moufang \(n\)-gon, for some \(n\). NEWLINENEWLINENEWLINETwo new characterization theorems for finite Moufang polygons, one purely combinatorial, the other group-theoretical, are proved. The first one says that every thick finite \(2\)-Moufang polygon is a \(3\)-Moufang polygon, and hence a Moufang polygon.