Proposition 7

(i) If $t \in \mathbf{Z}$, a t-th power mapping coincides on a neighbourhood of e with the mapping $g \mapsto g^t$.

(ii) The tangent mapping at e to a t-th power mapping is the homothety of ratio t.

(iii) If h is a t-th power mapping and h' a t'-th power mapping of G, $h \circ h'$ is a (tt')-th power mapping and $g \mapsto h(g)h'(g)$ is a (t + t')-th power mapping.

(iv) If h is a t-th power mapping and $u \in U^n(G)$, then $h_*(u) = t^n u$.

It suffices to prove the proposition when G is the Lie group germ defined by a complete normed Lie algebra and when the t-th power mappings considered are constructed using the exponential mapping $\phi = \operatorname{Id}_G$. But in that case everything is obvious.