For a Finite Group , let be the Subgroup generated by all the Sylow
*p*-Subgroup of . If is a projective curve in characteristic , and if , ..., are points of
(for ), then a Necessary and Sufficient condition that occur as the Galois Group of a finite
covering of , branched only at the points , ..., , is that the Quotient Group has
generators.

Raynaud (1994) solved the Abhyankar problem in the crucial case of the affine line (i.e., the projective line with a point deleted), and Harbater (1994) proved the full Abhyankar conjecture by building upon this special solution.

**References**

Abhyankar, S. ``Coverings of Algebraic Curves.'' *Amer. J. Math.* **79**, 825-856, 1957.

Harbater, D. ``Abhyankar's Conjecture on Galois Groups Over Curves.'' *Invent. Math.* **117**, 1-25, 1994.

Raynaud, M. ``Revêtements de la droite affine en caractéristique et conjecture d'Abhyankar.'' *Invent. Math.* **116**,
425-462, 1994.

© 1996-9

1999-05-25