A Note on Galois Modules and the Algebraic Fundamental Group of Projective Curves

Let X be a smooth projective connected curve of genus g ≥ 2 defined over an algebraically closed field k of characteristic p > 0. Let G be a finite group, P a Sylow p-subgroup of G and NG(P ) its normalizer in G. We show that if there exists an étale Galois cover Y → X with group NG(P ), then G is the Galois group wan étale Galois cover Y → X , where the genus of X depends on the order of G, the number of Sylow p-subgroups of G and g. Suppose that G is an extension of a group H of order prime to p by a p-group P and X is defined over a finite field Fq large enough to contain the |H|-th roots of unity. We show that integral idempotent relations in the group ring C[H] imply similar relations among the corresponding generalized Hasse-Witt invariants.