Example

Take $G =\mathbf{U}(n,\mathbf{C})$. We shall see later that the subgroup of diagonal matrices in G is a maximal torus of G and that its normalizer is the set of monomial matrices (Algebra, Chap. II, §10, no. 7, Example II) in G.

We conclude that, if $\Phi$ is a separating positive hermitian form on a finite dimensional complex vector space V and $\Gamma$ is a nilpotent subgroup of $\mathbf{U}(\Phi )$, there exists a basis of V for which the matrices of the elements of $\Gamma$ are monomial (“Blichtfeldt’s theorem”).