Exercise 7, § 2

Let $G$ be a topological group, $H$ a normal subgroup isomorphic to $\mathbf{R}^p \times \mathbf{T}^q$ ($p \geq 0,\ q \geq 0$), and suppose that there exists a surjective continuous homomorphism $\varphi : \mathbf{R} \to G/H$. Show that there exists a continuous homomorphism $f : \mathbf{R} \to G$ such that $f(\mathbf{R}) \cap H = \{e\}$ and $G = H \cdot f(\mathbf{R})$. [Using Exercise 4 b) and Chapter V, § 3, Exercise 8, reduce to the case where $G$ is not commutative and then, using Exercise 5, to the case $q = 0$; now use induction on $p$, together with Exercise 6.]