On uniqueness of automorphisms groups of Riemann surfaces

Let γ, r, s, ≥ 1 be non-negative integers. If p is a prime sufficiently large relative to the values γ, r and s, then a group H of conformal automorphisms of a closed Riemann surface S of order ps so that S/H has signature (γ, r) is the unique such subgroup in Aut(S). Explicit sharp lower bounds for p in the case (γ, r, s) ∈ {(1, 2, 1), (0, 4, 1)} are provided. Some consequences are also derived.