Opgave 8a. Show that an abelian group \(M\) has a unique \(\Bbb Z\)-module structure.
Bewijs. (\(\Leftarrow\)) If \(M\) is a \(\Bbb Z\)-module, then \(M\) is an abelian group by definition.

(\(\Rightarrow\)) Say \(A\) is an abelian group. We want to show that \(A\) is \(\Bbb Z\)-module. Define \(na = a^n = a + ... +a\) for \(n\in \Bbb Z\) and \(a \in A\). Then \[\begin{aligned} &(n+m)a = a^{n+m}=a^n+a^m= na + nm \\ &(nm)a = a^{nm} = (a^{m})^n = n(ma) \\ &n(a+b) = (a+b)^n = a^n + b^n = na + nb \quad \text{$A$ is abelian} \\ &1(a) = a^1 = a\end{aligned}\]