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Is it true that every finitely generated nilpotent group of matrices over $\mathbb C$ is conjugated to a subgroup of the upper triangular group?

If yes, what is a reference for that?

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    Lie-Kolchin Theorem2012-08-04
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    Since the Lie-Kolchin theorem applies to connected groups only it is useless here. (At least the classical version of Lie-Kolchin...)2012-08-05
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    Btw, there is a version of Lie-Kolchin theorem for linear unipotent groups. But the elements of a linear nilpotent group do not have to be unipotent.2012-08-05
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    I think torsion might give you trouble, but maybe not: over $\mathbb{C}$ you have the Jordan form.2012-08-05

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