In mathematics, a Godeaux surface is one of the surfaces of general type introduced by Lucien Godeaux in 1931.
Other surfaces constructed in a similar way with the same Hodge numbers are also sometimes called Godeaux surfaces.
The cyclic group of order 5 acts freely on the Fermat surface of points (w : x : y : z) in P3 satisfying w5 + x5 + y5 + z5 = 0 by mapping (w : x : y : z) to (w:ρx:ρ2y:ρ3z) where ρ is a fifth root of 1.
The quotient by this action is the original Godeaux surface.
The fundamental group (of the original Godeaux surface) is cyclic of order 5.