Exercise 2, § 4

Let G and H be two Lie groups and $f$ a Lie group morphism of G into H. (i) The kernel N of $f$ is a normal Lie quasi-subgroup of G and $L(N) = \mathrm{Ker}\ L(f)$. (Use exponential mappings of G and H). (ii) Let $g : G/N \to f(G)$ be the mapping derived from $f$ when passing to the quotient. If $f$ and $L(f)$ have closed images, and the topology of G admits a countable base, then $f(G)$ is a Lie quasi-subgroup of H with Lie algebra $\mathrm{Im}\ L(f)$ and $g$ is a Lie group isomorphism, where $G/N$ has the structure defined in Exercise 1. (Reduce it by means of (i) to the case where $N = \{e\}$.)