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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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    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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I realized that the answer to my question is NO. Take the quaternion group, realized by $\pm I, \pm \left(\begin{matrix} i & 0 \\ 0 & -i \end{matrix}\right), \pm \left(\begin{matrix} 0 & 1 \\ -1 & 0 \end{matrix}\right), \pm \left(\begin{matrix} 0 & i \\ i & 0 \end{matrix}\right).$ It is nilpotent but not triangularizable.

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    I$n$deed, a finite triangulizable group over a field of characteristic 0 should be abelian, since a unipotent element will have infinite order, and the only thing left are the diagonal matrices.2012-08-06