Summary: We study nilpotent groups that act faithfully on complex algebraic varieties. In the finite case, we show that when \(\mathbf{k}\) is a number field, a finite \(p\)-subgroup of the group of polynomial automorphisms of \(\mathbf{k}^d\) is isomorphic to a subgroup of \(\mathrm{GL}_d(\mathbf{k})\). In the case of infinite nilpotent group actions, we show that a finitely generated nilpotent group \(H\) acting on a complex quasiprojective variety \(X\) of dimension \(d\) can be embedded in a \(p\)-adic Lie group that acts faithfully and analytically on \(\mathbf{Z}_p^d\). As a consequence, we show that the virtual derived length of \(H\) is at most the dimension of \(X\).