Exercise 12, § 5

Let $p$ be a prime number. Show that the following conditions are equivalent:

(i) $p$ is not a torsion prime number of G;

(ii) for every subgroup F of G isomorphic to $(\mathbf{Z}/p\mathbf{Z})^n$ (for $n\in \mathbf{N}$), the centralizer of F in G is connected;

(ii$'$) for every subgroup F of G isomorphic to $(\mathbf{Z}/p\mathbf{Z})^2$, the centralizer of F in G is connected;

(iii) every subgroup of G isomorphic to $(\mathbf{Z}/p\mathbf{Z})^n$ for some integer $n$ is contained in a maximal torus;

(iii$'$) every subgroup of G isomorphic to $(\mathbf{Z}/p\mathbf{Z})^3$ is contained in a maximal torus.

(To prove that (i) $=\Rightarrow$ (ii), use Exerc. $11e$); to prove that (iii) $=\Rightarrow$ (i), use Exerc. $11f)$ and Exerc. 7.)