Abstract. We determine the values of q for which PSL2(q), acting on the cosets of a subgroup isomorphic to A4, S4 or A5, has monodromy genus 0.

Introduction

The monodromy genus γ(G; Ω) of a finite transitive permutation group (G, Ω) is the least genus of any branched covering S of the Riemann sphere with monodromy group (G, Ω); the monodromy genus γ(G) of a finite group G is the minimum of γ(G; Ω) as Ω ranges over all faithful transitive representations of G, that is, the least genus of any S with monodromy group isomorphic to G.

It is well-known [7] that cyclic, dihedral, alternating and symmetric groups all have genus 0. On the other hand, Guralnick and Thompson [3] have conjectured that at most finitely many simple groups of Lie type can be composition factors of monodromy groups of any given genus; Liebeck and Saxl [8] have verified this for simple groups of bounded Lie rank, but the general problem remains open.

In [5] I determined those q for which the natural representation of G = PSL2(q) on the projective line Ω = PG1(q) has genus 0 (all the prime-powers q ≤ 43 except 23, 27, 31 and 32). It is hoped eventually to determine all q for which PSL2(q) has genus 0 (in any representation), and to do this it is sufficient to consider the primitive representations, those for which the stabilisers H = Gα(α ∈ Ω) are maximal subgroups.