Exercise 7, § 5

Let $p$ be a prime number. Show that the following conditions are equivalent: (i) $p$ does not divide the order of the torsion subgroup of $\pi_1(G)$;

(ii) for every element $g$ of G of order $p$, the centralizer of $g$ in G is connected; (iii) every subgroup of G isomorphic to $(\mathbf{Z}/p\mathbf{Z})^2$ is contained in a maximal torus.

(To prove that (i) $=\Rightarrow$ (ii), use Exerc. $3c$); to prove that (iii) $=\Rightarrow$ (i), use Exerc. 6.)