Proposition 5

Let G be a connected topological group admitting a discrete normal subgroup D such that K = G/D is compact, and such that the commutator subgroup of K is dense in K. Then D is finite and G is compact.

The group G is locally isomorphic to K (GT, III, §2, No. 6, Prop. 19), hence is locally compact; since it is connected, it is generated by a compact neighborhood of e. By Lemmas 3 and 4, D is a finitely generated abelian group, hence is isomorphic to a group $\mathbf{Z}^r \times D_1$ with D_1 finite (A, VII, §4, No. 7, Th. 3). Suppose that r > 0. Then there exists a representation f of D onto $\mathbf{Z}$. By Prop. 4, f may be extended to a continuous representation g of G in R. By passage to quotients, g defines a continuous representation $g'$ of $K$ in $\mathbf{R}/\mathbf{Z}$; since $\mathbf{R}/\mathbf{Z}$ is abelian, the kernel of $g'$ contains the commutator subgroup of $K$, therefore $g'$ is trivial; in other words, $g(G) \subset \mathbf{Z}$. Since $G$ is connected, it follows that $g(G) = \{0\}$, which is absurd since $f(D) = \mathbf{Z}$. Thus $r = 0$ and $D$ is finite. Consequently $G$ is compact (GT, III, §4, No. 1, Cor. 2 of Prop. 2).