It is shown that if a simple group G acts conformally on a hyperbolic surface of least area (or alternatively, on a Riemann surface of least genus σ > 2), then G is normal in Aut(5) and the map Aut(*S) —> Aut((j) induced by conjugation is injective. For the preponderance of these minimal actions the group Aut(S)/G is isomorphic to a subgroup of Σ 3. It is shown how to compute Aut(S) purely in terms of the group-theoretic structure of G, in these cases. As examples and as part of the proof, the minimal actions and the groups Aut(iS) are completely worked out for A5, SL3(3), Mn and 1. Introduction. If G is a finite group, then G can act as a group of conformal (i.e. biholomorphic) automorphisms of a closed Riemann surface for infinitely many genera. Several authors, [C], [G-Sl], [G-S2], [HI], [M], [T] and [W], have considered the question of determining the least genus of a surface on which a given group can act conformally. Tucker [T] calls this least genus the strong symmetric genus of the group, though we will adopt the terminology of H. Glover and call this least genus the action genus. Actions on such surfaces we shall call genus actions. Conder [C] has determined the action genera of all the alternating groups. Glover and Sjerve [G-Sl], [G-S2] have determined the action genera for PSI^j?^), p a prime. Harvey [HI] and McLachlan [M] have worked out procedures to easily determine genus actions of cyclic and abelian groups, respectively. If the surface S has genus σ > 2, then Aut(S) is finite and, according to Hurwitz's famous theorem, |Aut(S)| 2.