Exercise 26, § 2

For every $\mathbf{Z}$-module $G$ and every integer $n > 0$, let $_n G$ denote the kernel of the endomorphism $x \mapsto nx$ of $G$. If $0 \to G' \to G \to G'' \to 0$ is an exact sequence, define a canonical homomorphism $d : _n G'' \to G'/nG'$ such that the sequence

is exact. If $_n G$ and $G/nG$ are finite, we write

then show that $_n G'$, $_n G''$, $G'/nG'$, $G''/nG''$ are finite and that