Exercise 6, § 1

a) Let $A$ be an abelian group of finite type, and $T$ a finite subset of $A$ not containing 0. There exists a subgroup $H$ of $A$ of finite index such that $H \cap T = \varnothing$. (Let $t \in T$. By using the structure of abelian groups of finite type, construct a subgroup of $A$ of finite index that does not contain $t$. Now proceed by induction on $\mathrm{Card}(T)$.)

b) Let $R$ be a root system and $P$ a closed symmetric subset of $R$. There exists a subgroup $H$ of $Q(R)$ of finite index such that $P = H \cap R$. (Pass to the quotient by the subgroup of $Q(R)$ generated by $P$, and use a) and Prop. 23.)