Proposition 22

Let $H$ be a Lie group, $U$ a Lie subgroup germ of $H$ and $N$ the set of $g \in H$ such that $U$ and $gUg^{-1}$ have the same germ at $e$ (General Topology, Chapter I, § 6, no. 10). Then $N$ is a subgroup of $H$ containing $U$. There exists one and only one analytic manifold structure on $N$ with the following properties:

(i) $N$ with this structure is a Lie group; (ii) $U$ is an open submanifold of $N$; (iii) the canonical injection of $N$ into $H$ is an immersion.

Clearly $N$ is a subgroup of $H$. If $g \in U$, then $ge \in U$ and $geg^{-1} \in U$, hence $gu \in U$ and $gug^{-1} \in U$ for $u$ sufficiently close to $e$ in $U$ and hence the germ of $gUg^{-1}$ at $e$ is contained in that of $U$; exchanging $g$ and $g^{-1}$, we see that the germs of $gUg^{-1}$ and $U$ at $e$ are equal. Hence $U \subset N$.

Let $V$ be an open neighbourhood of $e$ in $U$ such that $V = V^{-1}, V^2 \subset U$. Conditions (i), (ii), (iii) of Proposition 18 of no. 9 (where $G$ is replaced by $N$) are satisfied. Hence there exists an analytic manifold structure on $N$ with the following properties: ($\alpha$) $N$ with this structure is a Lie group; ($\beta$) $V$ is open in $N$; ($\gamma$) the manifold structures on $N$ and $U$ induce the same structure on $V$. Since $V$ is a submanifold of $H$, the canonical injection of $N$ into $H$ is an immersion at $e$ and hence at every point of $N$. Let $u \in U$. There exists an open neighbourhood V' of e in V such that the mapping $v \mapsto uv$ is an analytic isomorphism of V' onto an open neighbourhood of u in U (Proposition 20) and at the same time onto an open neighbourhood of u in N. Hence U is open in N and the identity mapping of U is an isomorphism with the given manifold structure on U and the open submanifold structure on N; in other words, U is an open submanifold of N.

Finally, we consider an analytic manifold structure on N with properties (i) and (ii) of the proposition and let N* be the Lie group thus obtained. Then the identity mapping of N into N* is étale at e and hence a Lie group isomorphism. This proves the uniqueness assertion of the proposition.

Let H be a Lie group, U a Lie quasi-subgroup germ of H and N the set of $g \in H$ such that U and $gUg^{-1}$ have the same germ at e. If K is of characteristic 0, there exists on G one and only one manifold structure with properties (i) and (ii) of Proposition 22. The proof is the same as for Proposition 22.