Proposition 6

The unique group homomorphism

such that $\lambda (f) =\gamma_o(f)$ for $f\in G_o$ and $\lambda ([k]) =\gamma (c_k)$ for $k\in K$ is surjective. Its kernel is the smallest normal subgroup of $G_o*F(K)$ containing the following elements:

($R_2$)$r_2(k, f) = [k]^{-1}f[k](c^{-1}_kk^*(f)^{-1}c_k)$

for $k\in K$ and $f\in G_o$;

($R'_3$)$r'_3(k, a) = [k](c^{-1}_kk^*(f_{(a,k)})^{-1}f_{(a,k)}))$

for $k\in K-\{e\}$ and $a\in A_k$

($R''_3$)$r''_3(k, h) = [kh]^{-1}[k][h](c^{-1}_hh^*(c^{-1}_k)c_{kh})$

for $k$ and $h\in K$.

Since G is transitive, the mapping which to an element $k\in K$ associates the orbit of $(o, k)$ is a bijection of K onto the set of orbits of H. Thus Orb(G) is identified with $\{o\}$ and Orb(H) with K. The framework $\Gamma_0$ of the pair $(\varphi , \psi )$ is then identified with the quiver having a unique vertex $o$ and whose set of arrows is K. Let T be the subquiver of $\Gamma_0$ with set of vertices $\{o\}$ and whose set of arrows is empty; the associated graph is the unique maximal tree of the graph $\widetilde{\Gamma}_0$.

The family $(o,(o, k)_{k\in K},(e_o)_{k\in K},(c_k)_{k\in K},T, o)$ is a basic equipment of the pair $(\varphi , \psi )$.

The mapping $f\mapsto (f, k)$ defines an isomorphism of the isotropy group $G_o$ onto the isotropy group $H_{(o,k)}$; by means of this isomorphism, the homomorphisms $\varphi_k$ and $\psi_k$ of $H_{(o,k)}$ into $G_o$ defined by Prop. 5 of II, p. 208 are given by

(3) $\varphi_k(f, k) =f$ et $\psi_k(f, k) =c^{-1}_kk^*(f)c_k$,

for $k\in K$ and $f\in G_o$.

The quiver $H\times_GH$ is the equalizer in $H\times H$ of the quiver morphisms $\psi \circ$ pr$_1$ and $\varphi \circ$ pr$_2$. Its vertices are the pairs $((a, k),(b, h))$ where $a$ and $b$ are vertices of G and $k$ and $h$ elements of K such that $b=k^*a$, and its arrows are the pairs $((f, k),(g, h))$ where $f$ and $g$ are arrows of G, and $k$ and $h$ elements of K such that $g=k^*f$; the origin of the arrow $((f, k),(g, h))$ is equal to $((o(f), k),(o(g), h))$; its end is equal to $((t(f), k),(t(g), h))$.

One then defines a quiver morphism $\mu$ from $H\times_GH$ into H by putting $\mu((a, k),(b, h)) = (a, kh)$ and $\mu((f, k),(g, h)) = (f, kh)$. We have $\varphi \circ \mu=\varphi \circ$ pr$_1$ and $\psi \circ \mu=\psi \circ$ pr$_2$.

Let $x$ and $y$ be vertices of H such that $\varphi (x) =\varphi (y)$. Then there exist a vertex $a$ of G and elements $k$ and $h$ of K such that $x= (a, k)$ and $y= (a, h)$. Put $z= (h^*a, h^{-1}k)$; then $\mu(y, z) =x$. This shows that the pair $(\varphi , \psi )$ satisfies the hypotheses of Prop. 4 (II, p. 202).

A vertex $(a, k)$ of H belong to the groupoid Ker($\varphi , \psi$ ), equalizer of $\varphi$ and $\psi$, if and only if $k^*a=a$, that is to say, if $k$ belong to the stabilizer of the vertex $a$ in the group K. Let $A_1$ be the set of pairs $(a, k)$, for $k\in K$ and $a\in A_k$; it meets all the orbits of the subgroupoid Ker($\varphi , \psi$ ) of H. Let $Z_1$ be the subset of $\Omega (\widetilde{\Gamma})$ formed by the sequences of the form $(((a, k),1))$, for $(a, k)\in A_1$. For $\mathbf{z}= ((a, k),1)\in Z_1,(f_{(a,k)}, k)$ is an arrow of H joining $(a, k)$ to $(o, k)$. Put $f(\mathbf{z}) = ((f_{(a,k)}, k))$.

The set $A_2$ of vertices of $H\times_GH$ of the form $((o, k),(k^*o, h))$, for $(k, h)\in K^2$, meets all the orbits of $H\times_GH$. Observe also that one has $\mu((o, k),(k^*o, h)) = (o, kh)$. The arrows $(e_o, k)$, $(c_k, h)$, $(e_o, kh)$ in H join respectively $(o, k)$, $(k^*o, h)$, $(o, kh)$ to $(o, k)$, $(o, h)$ and $(o, kh)$. Let $Z_2$ denote the set of sequences of the form $(((o, k),1),((k^*o, h),1),((o, kh),-1))$ in $\Omega (\widetilde{\Gamma})$; for such an element $\mathbf{z}$ of $Z_2$, put $f(\mathbf{z}) = ((e_o, k),(c_k, h),(e_o, kh))$.

By Prop. 4 of II, p. 202, the set $Z = Z_1\cup Z_2$ and the family $(f(\mathbf{z}))_{z\in Z}$ is a complementary equipment.

Let $k\in K$ and let $a\in A_k$. The element $C((a, k),1)$ of the group $G_o*F(K)$ defined by formula (2) of II, p. 208 is equal to

Let $k$ and $h\in K$ be given. One verifies that the element

of the group $G_o*F(K)$ defined by formula (2) of II, p. 208 is equal to

The elements $r'_3(e, a)$ given by the relations ($R'_3$) for $k=e$ are all equal to $[e]c^{-1}_e$, an element of the group $G_o*F(K)$ which one obtains by applying the relation ($R''_3$) to $k=h=e$. Taking account of relations (3), (4), and (5), the proposition therefore follows from II, p. 208, Prop. 5.