On the order of automorphism groups of Klein surfaces

Abstract

A problem of special interest in the study of automorphism groups of surfaces are the bounds of the orders of the groups as a function of the genus of the surface.May has proved that a Klein surface with boundary of algebraic genus p has at most 12(p–1) automorphisms [9].In this paper we study the highest possible prime order for a group of automorphisms of a Klein surface. This problem was solved for Riemann surfaces by Moore in [10]. We shall use his results for studying the Klein surfaces that are not Riemann surfaces. The more general result that we obtain is the following: if X is a Klein surface of algebraic genus p, and G is a group of automorphisms of X, of prime order n, then n ≤ p + 1.