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We prove a version of the Hilbert Irreducibility Theorem for linear algebraic groups. Given a connected linear algebraic group , an affine variety and a finite map , all defined over a finitely generated field of characteristic zero, Theorem 1.6 provides the natural necessary and sufficient condition under which the set contains a Zariski dense sub-semigroup ; namely, there must exist an unramified covering and a map such that . In the case , is the additive group, we reobtain the...
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