Corollary 1

Suppose further that, for every arrow $f$ of F, we have $o(f) =t(f)$. Then the origin and the end of every path in the quiver Γ coincide. Let $a$ be a vertex of G and let A be its orbit in G. For every element $c$ of F whose origin $b$ belongs to A, let us fix an arrow $f_c$ of G joining $a$ to $b$ and put $\kappa (c) =$ Int($f_c$)$(\delta (c)) =$ $f_c\delta (c)f_c^{-1}$.

The group $H_a$ is then the distinguished subgroup of $G_a$ generated by the arrows $\kappa (c)$.

By Prop. 8, the arrows $\kappa (c)$ are contained in the group $H_a$. Let $N_a$ denote the smallest normal subgroup of $G_a$ containing them. By that proposition, it is enough to prove that, for every vertex $b$ of G belonging to A, every arrow $f_{ab}$ joining $a$ to $b$ in G, and every element $c$ of Grp(Γ)$_b$, Int($f_{ab}$)$(\delta (c))$ belongs to $N_a$. Since the origin and the term of every element of F are equal, a loop at a vertex $b$ in the graph $\widetilde{\Gamma}$ is of the form $(b,(f_1, \varepsilon_1), b, . . . ,(f_n, \varepsilon_n), b)$, where for every $i\in \{1, . . . , n\},f_i$ is an element of F with origin $b$ and $\varepsilon_i\in \{-1,1\}$. We have

Int($f_{ab}$)$(\delta (c)) =f_{ab}f_c^{-1}\kappa (f_1)^{\varepsilon_1}. . . \kappa (f_n)^{\varepsilon_n}f_cf_{ab}^{-1}$.

Since $f_{ab}f_c^{-1}\in G_a$, it follows that Int($f_{ab}$)$(\delta (c))$ belongs to $N_a$.