2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/160359Let X be a smooth projective algebraic curve of genus g minus $r\geq 1$ points defined over an algebraically closed field k of characteristic $p\geq 0$. The structure of the largest prime to p quotient of the étale fundamental group is well known by transcendental methods : it is isomorphic to the largest prime to p quotient of a free pro-finite group on 2g+r-1 generators. We show that, with purely algebraic means, we can prove the corresponding result for the largest pro-solvable quotient of these groups.v2, minor changes, exposition improvedAlgebraic GeometryNumber Theorytext