Corollary

Let Z be a subset of $\Omega (\widetilde{\Gamma})$; in order that the conjugacy classes $c(\mathbf{z})$, for $\mathbf{z}\in$ Z, generate the kernel of the canonical homomorphism $\beta_{a_0}: G_{a_0}\rightarrow G'_{\beta(a_0)}$, it is enough that the smallest distinguished subset of $\Omega (\widetilde{\Gamma})$ containing Z be equal to $\Omega (\widetilde{\Gamma})$.

Let N be the smallest subgroup of $G_{a_0}$ containing $c(\mathbf{z})$ for every $\mathbf{z}\in Z$; it is distinguished. Let $Z'$ be the set of elements $\mathbf{z}$ of $\Omega (\widetilde{\Gamma})$ such that $c(\mathbf{z})\in N$. By proposition 2, $Z'$ is a distinguished subset of $\Omega (\widetilde{\Gamma})$. It contains Z. By hypothesis, it is therefore equal to $\Omega (\widetilde{\Gamma})$. It then follows from prop. 1 (II, p. 197) that N is the kernel of the homomorphism $\beta_a$.