# Abhyankar's conjecture

From Groupprops

This article is about a conjecture in the following area in/related to group theory: algebraic geometry. View all conjectures and open problems

*This article or section of article is sourced from*:Mathworld

## Contents

## Statement

### Given data

A finite group , a prime number , a nonsingular projective curve , defined over a field of characteristic . (where ) are points of . Let denote the genus of .

### Statement

Let denote the subgroup generated by all the -Sylow subgroups of . Then, the following are equivalent:

- occurs as the Galois group of a branched covering of , branched only at the points
- The quotient group has generators.

## Progress towards the conjecture

Raynaud settled the conjecture in the affine case, and Harbater proved the full conjecture by building upon this special solution.