Exercise 4, § 4

(a) Let K = $\mathbf{R}$ or $\mathbf{C}$ or an ultrametric field with residue field of characteristic > 0. Let G be a Lie group over K, U an open neighbourhood of 0 in L(G) and $\phi : U \to G$ a mapping which is differentiable at 0 and such that $\phi(0) = e,\ T_0(\phi) = \mathrm{Id}_{L(G)}$ and $\phi((\lambda + \lambda')b) = \phi(\lambda b)\phi(\lambda'b)$ for $\lambda b,\ \lambda'b,\ (\lambda + \lambda')b$ in $U$. Then $\phi$ coincides on a neighbourhood of 0 with an exponential mapping.

(b) Let $k$ be a field of characteristic 0. Let $K$ be the valued field $k((X))$, which is of characteristic 0. Let $G$ be the additive Lie group $k[[X]]$. Let $\phi$ be the continuous $k$-linear mapping of $G$ into $G$ such that $\phi(X^n) = X^n + X^{2n}$ for every integer $n \geqslant 0$. Then $\phi$ satisfies the conditions of (a), but coincides on no neighbourhood of 0 with an exponential mapping.