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Let $A$ be a finitely generated abelian group and $f:A\rightarrow A$ a surjective homomorphism. How do I prove that $f$ is an isomorphism?

And if $f$ were injective instead of surjective would the statement still hold?

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    $\begingroup$ Injectivity does not imply surjectivity: Just take $f : \mathbb{Z} \to \mathbb{Z}$ given by $x \mapsto 2x$. $\endgroup$ – Prahlad Vaidyanathan Nov 27 '15 at 16:04
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If $A$ is a finitely generated abelian group, every subgroup of $A$ is finitely generated (because $A$ is abelian, see proof here). So $A$ is Noetherian as a $\Bbb Z$-module. For Noetherian modules, every epimorphism is an isomorphism, see proof here.

We're using that a module is Noetherian if and only if every submodule is finitely generated. The analogous result for injectivity is that for Artinian modules, every monomorphism is an isomorphism.

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  • $\begingroup$ Clearly finitely generated is necessary, but I could not come up with a counter example. Of course I thought of something like $A=\bigoplus \mathbb{Z}$ and sending $e_i$ to $e_{i+1}$ or something, but its not surjective. Do you know an example? $\endgroup$ – Rico Nov 27 '15 at 16:59
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You might have already seen the solution after two years now, but I shall provide you with one purely group-theoretical, without noetherian modules.

  1. It's a theorem that every finitely generated abelian group is residually finite. One of the equivalent characterisations of residual finteness is that for every non trivial element of the group there is a normal subgroup of finite index not containing that element. It is trivial to prove the residual finiteness for finitely generated abelian groups, since they are direct products of cyclic groups, which are indeed residually finite and residual finiteness is preserved under direct products.

  2. It's also a well known theorem/exercise that every finitely generated residually finite group is Hopfian, namely that every epimorphism of the group is also an isomorphism.

In order to prove the 1,2 you will need the equivalent characterisations of residual finiteness, which you can find in every good advanced algebra book (e.g. Rotman).

From 1,2 it follows that every finitely generated abelian group is Hopfian, which is what you wanted to prove.

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