Exercise 13, § 11

Denote by $\mathscr{C}$ (resp. $\mathscr{D}$) the set of classes of $\mathbf{Z}$-modules of finite length (resp. finitely generated $\mathbf{Z}$-modules). Show that the group homomorphism $\gamma_{\mathscr{D},\mathscr{C}}$ from $K(\mathscr{C})$ to $K(\mathscr{D})$ is zero.