Exercise 10, § 5

Let R be a root system and $p$ a prime number. Show that the following properties are equivalent:

(i) $p$ is a torsion prime number of an irreducible component of R (Exerc. $9b$)$)$; (ii) there exists a symmetric closed subset $R_1$ of R, distinct from R and maximal for these properties, such that $(Q(R^{\vee}) : Q(R^{\vee}_1)) =p(Q(R^{\vee})$ denotes the $\mathbf{Z}$-module generated by the inverse roots of R and $Q(R^{\vee}_1)$ the submodule generated by the $\alpha^{\vee}$ for $\alpha \in R_1)$;

(iii) there exists a symmetric closed subset $R_1$ of R such that the $p$-torsion submodule of $Q(R^{\vee})/Q(R^{\vee}_1)$ is non-zero.

(To prove that (i) $=\Rightarrow$ (ii), use Exerc. 4 of Chap. VI, §4; to prove that (iii) $=\Rightarrow$ (i), use Exerc. $9d$)$.)$

In this case $p$ is said to be a torsion prime number of R.