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I'm trying to figure out what common group (if any!) the group $F / F_{fin}$ is isomorphic to, where $F = \{f : \mathbb Z \to \mathbb Z\}$ (with pointwise addition, so $F$ is abelian) and $F_{fin} = \{f \in F : \{x \in \mathbb Z : f(x) \neq 0\} \text{ is finite}\}$. I thought to use the First Isomorphism Theorem, but I can't seem to think of a homomorphism from $F$ with kernel $F_{fin}$. If we replace $\mathbb Z$ with a finite group, obviously the quotient would be trivial, since every function would have finite support. What happens in this case, when the base group ($\mathbb Z$) is infinite?

Apologies if this is really simple.

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    It's not a familiar group. – 2017-02-22
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    @QiaochuYuan does it have a name? Where can I learn more about it? – 2017-02-22
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    Your group $F$ is just the [*Baer-Specker group*](https://en.wikipedia.org/wiki/Baer%E2%80%93Specker_group) $\prod\limits_{\mathbb{N}}\mathbb{Z}$. With regards to the subgroup $\bigoplus\limits_{\mathbb{N}}\mathbb{Z}$, you may find [this discussion interesting](http://math.stackexchange.com/questions/333206/countable-infinite-direct-product-of-mathbbz-modulo-countable-direct-sum). – 2017-02-22
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    @ZevChonoles do you know any reference about the quotient $F/F_{fin}$? In the question you link the OP mentions only that they don't know much about it. – 2017-02-22

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