Abhyankar's Conjecture
For a Finite Group 
, let 
be the Subgroup generated by all the Sylowp-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 finitecovering 
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 pointdeleted), and Harbater (1994) proved the full Abhyankar conjecture by building upon this special solution.
See also Finite Group, Galois Group, Quotient Group, Sylow p-Subgroup
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.
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