Exercise 13, § 6

Let G be a real Lie group, $L = L(G)$, and $\phi : L \to G$ a mapping which is differentiable at 0, such that $T_0(\phi) = \mathrm{Id}_L$ and such that $\phi(nx) = \phi(x)^n$ for all $x \in L$ and $n \in \mathbf{Z}$. Then $\phi = \exp_G$. (Let V be a neighbourhood of 0 in L and W a neighbourhood of e in G such that $\theta = \exp_G | V$ is an analytic isomorphism of V onto W. Let $\psi = \theta^{-1} \circ (\phi | \phi^{-1}(W))$. Then $T_0(\psi) = \mathrm{Id}_L$, whence $\psi = \mathrm{Id}_L$ using the equality $\psi\left( \frac{1}{n} x \right) = \frac{1}{n} \psi(x)$ for x sufficiently close to 0 and $n \in \mathbf{Z} - \{0\}$.