Corollary 3

Suppose that the groupoid G is simply transitive. Let $K_0$ be the subgroup of K generated by the stabilizers of the vertices of G. The mapping from K into $(G/K)_{\gamma(o)}$ which to $k\in K$ associates $\gamma (c_k)$ is a surjective group homomorphism, with kernel $K_0$.

If an element $k\in K$ fixes a vertex $a$ of G, the element $g^{-1}kg$ fixes the vertex $g^*a$; this implies that $K_0$ is a normal subgroup of K.

By proposition 5, the unique homomorphism $\lambda : F(K)\rightarrow$ $(G/K)_{\gamma(o)}$ such that $\lambda ([k]) =\gamma (c_k)$ is surjective, and its kernel is the smallest normal subgroup of F(K) containing the elements $[k]$, where $k$ is an element of K which fixes a vertex of G, and the elements $[kh]^{-1}[k][h]$, where $(k, h)\in K^2$. In particular, the mapping $\lambda ': K\rightarrow (G/K)_{\gamma(o)}$, defined by $\lambda '(k) =\gamma (c_k) =\lambda ([k])$ for $k\in K$, is a group homomorphism. One has $\lambda =\lambda '\circ p$, where $p: F(K)\rightarrow K$ denotes the canonical surjective group homomorphism. Consequently, the homomorphism $\lambda '$ is surjective and its kernel is the smallest normal subgroup of K which contains the elements $k$, for $k$ fixing a vertex of G, that is to say $K_0$. This proves the proposition.