Corollary 1

Let $G'$ be a locally compact group, $G$ a finite-dimensional Lie group over $\mathbf{R}$ (resp. $\mathbf{Q}_p$) and $f$ a continuous morphism of $G'$ into $G$. If the kernel of $f$ is discrete, $G'$ is a finite-dimensional real (resp. $p$-adic) Lie group.

There exists a compact neighbourhood $V$ of $e$ in $G'$ such that $f|V$ is a homeomorphism of $V$ onto a compact subspace of $G$. If $U$ is a sufficiently small open neighbourhood of $e$ in $G$, the hypotheses of Theorem 2 are satisfied with $H = f(V) \cap U$. Hence $H$ is a Lie subgroup germ of $G$. Let $W$ be the inverse image of $H$ under $f|V$. Then $W$ is a neighbourhood of $e$ in $G'$. Let $W$ be given the analytic manifold structure transported from that on $H$ by $(f|W)^{-1}$. For all $z \in G'$, the mapping $x \mapsto f(z)x f(z)^{-1}$ of $G$ into $G$ is analytic; hence there exists an open neighbourhood $W'$ of $e$ in $W$ such that the mapping $x' \mapsto zx'z^{-1}$ of $W'$ into $W$ is analytic. By Proposition 18 of § 1, no. 9, there exists on $G'$ a Lie group structure which induces on a sufficiently small open neighbourhood of $e$ the same analytic structure as $W$ and hence the same topology as the initial topology on $G'$.