Realization conjecture for finite groups as Galois groups of coverings

Let k(x)k(x) be the function field of the projective line, and let EE be a finite regular Galois extension of k(x)k(x) with Galois group GG. Equivalently, the inclusion k(x)Ek(x)\hookrightarrow E defines a ramified Galois covering CP1C\to\mathbb{P}^{1} over kk with Galois group GG. Realization conjecture. Every finite group GG occurs as the Galois group of such a covering. This is a realization problem for finite groups as Galois groups of regular function-field extensions, and the supplied text does not state whether the claim is known or open.

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Primary source

Alberto Besana and Cristina Martinez, “Enumerative geometry of the curves defined by y^d=f(x)”, arXiv:1111.6456 (2012).

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