TITLE: Is every nilpotent linear group triangularizable?
QUESTION [5 upvotes]: 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?

REPLY [3 votes]: 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.