No finitely generated abelian group is divisible

Prove that no finitely generated abelian group is divisible.


Let A be a nontrivial divisible finitely generated abelian group. By FTFGAG, we have A \cong \mathbb{Z}^r \times B, where B is a finite abelian group. By this previous exercise, A is divisible if and only if \mathbb{Z}^r and B are divisible. By this previous exercise, if B is nontrivial then it is not divisible; thus B = 1. Again by this result, since r \geq 1, A is divisible if and only if \mathbb{Z} is divisible.

Note that there does not exist an element k \in \mathbb{Z} such that 2k = 1. Thus \mathbb{Z} is not divisible, hence A is not divisible, a contradiction. Thus no such group A exists.

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