Maximal group actions on compact oriented surfaces

Suppose S   is a compact oriented surface of genus σ≥2σ≥2 and CpCp is a group of orientation preserving automorphisms of S   of prime order p≥5p≥5. We show that there is always a finite supergroup G>CpG>Cp of orientation preserving automorphisms of S   except when the genus of S/CpS/Cp is minimal (or equivalently, when the number of fixed points of CpCp is maximal). Moreover, we exhibit an infinite sequence of genera within which any given action of CpCp on S   implies CpCp is contained in some finite supergroup and demonstrate for genera outside of this sequence the existence of at least one CpCp-action for which CpCp is not contained in any such finite supergroup (for sufficiently large σ).