Proposition 1

Let G, G' be topological groups and f a continuous morphism of G into G'. Suppose that one of the following three cases holds: (a) G is a real Lie group and G' is a p-adic Lie group; (b) G is a p-adic Lie group and G' is a real Lie group; (c) G is a p-adic Lie group and G' is a $p'$-adic Lie group with $p \neq p'$. Then f is locally constant.

Case (a). Let $G_0$ be the identity component of G. Then $f(G_0)$ is a connected subgroup of $G'$ and hence $f(G_0) = \{e\}$ and $G_0$ is open in G.

Case (b). Let $V'$ be a neighbourhood of $e$ in $G'$ such that every subgroup of $G'$ contained in $V'$ reduces to $\{e\}$ (§ 4, no. 2, Corollary 1 to Theorem 2). There exists a neighbourhood V of $e$ in G such that $f(V) \subset V'$. Then there exists an open subgroup $G_1$ of G such that $G_1 \subset V$ (§ 7, no. 1, Proposition 1). Then $f(G_1) = \{e\}$.

Case (c). By § 7, Theorem 4 and the Corollary to Proposition 8, there exists a neighbourhood $V'$ of $e$ in $G'$ such that, for all $x' \in V' - \{e\}$, ${x'}^{p^n}$ does not tend

† See for example D. Montgomery and L. Zippin, Topological transformation groups, Interscience tracts in pure and applied mathematics, no. 1, Interscience publishers, New York 1955 (in particular pp. 169 and 184).

to $e$ as $n$ tends to $+\infty$. There exists a neighbourhood $V$ of $e$ in $G$ such that $f(V) \subset V'$. By § 7, Theorem 4 and Proposition 9, there exists an open subgroup $G_1$ of $G$ such that $G_1 \subset V$ and such that, for all $x \in G_1$, $x^{p^n}$ tends to $e$ as $n$ tends to $+\infty$. Then $f(G_1) = \{e\}$.