91Ó°ÊÓ

(i) Prove that an abelian group \(G\) is finitely generated if and only if it is a quotient of a free abelian group of finite rank.(ii) Every subgroup \(H\) of a finitely generated abelian group \(G\) is itself finitely generated. Moreover, if \(G\) can be generated by \(r\) elements, then \(H\) can be generated by \(r\) or fewer elements.

If \(F\) is a free abelian group of finite rank \(n\), then a subgroup \(H\) of \(F\) has finite index if and only if \(H\) is free abelian of rank \(n\).

Let \(F\) be free abelian of rank \(n\) and let \(H\) be a subgroup of the same rank. Let \(\left\\{x_{1}, \ldots, x_{n}\right\\}\) be a basis of \(F\), let \(\left\\{y_{1}, \ldots, y_{n}\right\\}\) be a basis of \(H\), and let \(y_{j}=\sum m_{U} x_{i}\). Prove that $$ [F: H]=\left|\operatorname{det}\left[m_{i j}\right]\right| $$

If \(G\) and \(H\) are divisible groups each of which is isomorphic to a subgroup of the other, then \(G \cong H\). Is this true if we drop the adjective "divisible"?

Let \(A\) denote the dyadic rationals and let \(B\) denote the triadic rationals: $$ B=\left\\{q \in \mathbb{Q}: q=a / 3^{k}, a \in \mathbb{Z} \text { and } k \geq 0\right\\} \text { . } $$ Let \(G\) be the subgroup of \(Q \oplus \mathbb{Q}\) generated by $$ \\{(a, 0): a \in A\\} \cup\\{(0, b): b \in B\\} \cup\left\\{\left(\frac{1}{3}, \frac{1}{5}\right)\right\\} . $$ Prove that \(G\) is an indecomposable group of rank \(2 .\)