Proposition 11

Let $X$ be a Lie group and $G$ a Lie subgroup of $X$.

(i) There exists on the homogeneous set $X/G$ one and only one analytic manifold structure such that the canonical projection $\pi$ of $X$ onto $X/G$ is a submersion. The law of operation of $X$ on $X/G$ is analytic. For all $x \in X$, the kernel of $T_x(\pi)$ is obtained from $T_e(G)$ by $T_e(\gamma(x))$.

(ii) If $G$ is normal in $X$, $X/G$ is a Lie group with its group structure and the manifold structure defined in (i). The mapping $\pi$ is a Lie group morphism.

By General Topology, Chapter III, § 4, no. 1, Example 1, $G$ operates properly and freely on $X$ by right translation. Hence the first assertion of (i) follows from

Proposition 10 of no. 5. The second follows from the Remark of no. 5. Since $\pi$ is a submersion, the kernel of $T_x(\pi)$ is the tangent space at $x$ to

and hence is obtained from $T_e(G)$ by $T_e(\gamma(x))$.

Suppose that $G$ is normal. Let $m$ be the mapping $(x, y) \mapsto xy^{-1}$ of $(X/G) \times (X/G)$ into $X/G$. Then $(m \circ (\pi \times \pi))(x, y) = \pi(xy^{-1})$ for all $x, y$ in $X$. Hence $m \circ (\pi \times \pi)$ is analytic. As $\pi \times \pi$ is a surjective submersion, $m$ is analytic (Differentiable and Analytic Manifolds, R. 5.9.5), whence (ii).

The homogeneous set $X/G$ with the manifold structure defined in (i) is called the quotient (left) Lie homogeneous space of $X$ by $G$. The (right) Lie homogeneous space $G \setminus X$ is defined analogously. When $G$ is normal, the Lie group $X/G$ defined in (ii) is called the quotient Lie group of $X$ by $G$.