Hilbert Irreducibility Theorem over linear algebraic groups

Abstract
We present the following generalization of Hilbert Irreducibility Theorem: {\it Given a simply connected linear algebraic group G over a number field k, a Zariski-dense sub-semigroup $\Gamma \subset G(k)$ and a dominant map $\pi: V\to G$ from an algebraic variety $V$ over $k$, admitting no section, the image $\pi(V(k))$ does not contain $\Gamma$.} In the case where $k$ is the field of rational numbers, $G=G_a$ and $\Gamma$ is the semigroup of natural numbers, this statement is equivalent to Hilbert's original one.