Exercise 2, § 1

Let G be a topological group, such that the commutator subgroup of G is dense in G. Show that every continuous representation $\varphi$ of G in a Hausdorff abelian group satisfies $\varphi(G) = \{e\}$. From this, deduce that if G is locally compact, then every relatively invariant complex measure on G is invariant. In particular, G is unimodular.