The Flag-Transitive Collineation Groups of the Finite Desarguesian Affine Planes

Abstract

Let π be the Desarguesian affine plane of order n = p r , for p a prime and r a positive integer. A collineation group G of π is defined to be flag-transitive on π if G is transitive on the set of incident point-line pairs, or flags , of π . Further, G is doubly transitive on π if G is doubly transitive on the points of π . Clearly, G is flag transitive if G is doubly transitive on π . The purpose of the following study is the explicit determination of the flagtransitive and the doubly transitive collineation groups of π (I am indebted to D. G. Higman for suggesting this problem). The results can be summarized in Theorems 1′ and 2′ below (a complete description of the results is contained in Sections 12-15).