Unramified Abelian Extensions of Galois Covers

We consider a ramified Galois cover φ : X̂ → Px of the Riemann sphere Px, with monodromy group G. The monodromy group over Px of the maximal unramified abelian exponent n cover of X̂ is an extension nG̃ of G by the group (Z/nZ), where g is the genus of X̂. Denote the set of linear equivalence classes of divisors of degree k on X̂ by Pic(X̂) = Pic. This is equipped with a natural G action. We show that the equivalence class of the extension nG̃ → G is determined by the element of H(G, Pic) representing Pic (§2.2). From this we give an effective criterion (involving the Schur multiplier of G) to decide when this group extension splits for all n (§4.2). In particular we easily produce examples from this of cases where X̂ has G invariant divisor classes of degree 1, but no G invariant divisor of degree 1 (§5.1). The extension nG̃ → G naturally factors into a sequence nG̃ → H → G where H is the smallest quotient of nG̃ giving a frattini cover (§1.1) that fits between nG̃ and G. Extension of the main result of §4.2 would consider all maximal quotients M of nG̃ such that M → G splits. We note that the sequence including such an M factors through H, and by example we demonstrate that such maximal quotients M may not be unique (§5.2). INTRODUCTION: We consider a ramified Galois cover φ : X̂ → Px of the Riemann sphere Px, with monodromy group G. Choose an integer n > 1. Let X̂n be the maximal unramified abelian exponent n cover of X̂. The monodromy group Vn of this cover is isomorphic to (Z/nZ), where g is the genus of X̂. Composing with φ we get a Galois cover X̂n → Px, whose monodromy group is an extension of G by Vn: