Theorem 2

When K is ultrametric and locally compact and G is a compact Lie group, assertions (i), (ii), (iii), (v) and (vi) of Theorem 1 are true. (a) The uniqueness of the analytic structure considered in (i) is obvious. (b) Suppose that G is the Lie group defined by the normable Lie algebra L. Then G is an open and closed ball in L. Let $w \in \mathrm{Aut}\ G$. Then $L(w)$ coincides with $w$ in a neighbourhood of 0. Let $x \in G$. Let $p$ be the characteristic of the residue field. Then $p^n x$ tends to 0 as $n$ tends to $+\infty$. There therefore exists $n$ such that $w(p^n x) = L(w)(p^n x)$. Therefore

whence $w(x) = L(w)(x)$. Thus, $w = L(w)|G$.

Let $\Gamma$ be the set of $\gamma \in \mathrm{Aut}\ L(G)$ such that $\gamma(G) = G$. As G is open and compact in $L(G)$, $\Gamma$ is an open subgroup of $\mathrm{Aut}\ L(G)$. By the above, Aut G is identified with $\Gamma$, whence there is a Lie group structure on Aut G, with which properties (i), (ii), (iii) and (vi) of Theorem 1 are obvious. Property (v) follows from Propositions 1 and 3 of no. 1.

(c) We pass to the general case. By § 7, no. 1, Proposition 1, there exists an open compact subgroup G_0 of G which is of the type considered in (b). Then G is generated by G_0 and a finite number of elements x_1, x_2, ..., x_n. Let Aut_1G be the set of u ∈ Aut G such that u(G_0) = G_0 and u(x_iG_0) = x_iG_0 for 1 ≤ i ≤ n. We define as in the proof of Theorem 1, part (c), a semi-direct product P of Aut G_0 by G_0^n and an injective homomorphism ζ of Aut_1G into P, whose image is closed in P.

(d), (e): the argument is exactly as in parts (d), (e) of the proof of Theorem 1 with R replaced by Q_p and using Proposition 3 instead of Proposition 2.