Proposition 4

Let G be a finite-dimensional Lie group and H a subgroup of G. Let r \in \mathbf{N}_\mathbf{K}. The following conditions are equivalent: (i) H is an integral subgroup of G; (ii) with the Lie group structure induced by that on G, H is connected; (iii) H is C^r-connected. (iii) \Rightarrow (i): obvious. (i) \Rightarrow (iii): suppose that H has a Lie group structure such that H is an integral subgroup of G. Using the notation of Remark 3, the set of y \in H such that property P_{H, e, y} holds is an open subgroup of H. As H is connected, this subgroup is equal to H and hence condition (iii) is satisfied.

(iii) ⇒ (ii): suppose that condition (iii) is satisfied and let H be given the structure induced by the Lie group structure on G. Let h be the subalgebra tangent to H at e. The identity component H_0 of H is an integral subgroup of G such that L(H_0) = h. We show that H = H_0. It suffices to prove the following: let I be a connected open subset of K, f a mapping of class C^r of I into G such that f(I) ⊂ H and λ and μ two points of I; if f(λ) ∈ H_0, then f(μ) ∈ H_0. But, for all v ∈ I, (T_v f)(K) ⊂ f(v)h by definition of h, so that our assertion follows from Proposition 3.