What is $F / F_{fin}$ isomorphic to, where $F$ is the group of integer functions and $F_{fin}$ is the subgroup of finitely-supported functions?

Asked Feb 22 '17 at 01:10

Active Feb 22 '17 at 01:22

Viewed 38 times

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.

edited Feb 22 '17 at 01:22

asked Feb 22 '17 at 01:10

feralin

1,693

It's not a familiar group. – Qiaochu Yuan Feb 22 '17 at 01:31
@QiaochuYuan does it have a name? Where can I learn more about it? – feralin Feb 22 '17 at 01:32
2

Your group $F$ is just the Baer-Specker group $\prod\limits_{\mathbb{N}}\mathbb{Z}$. With regards to the subgroup $\bigoplus\limits_{\mathbb{N}}\mathbb{Z}$, you may find this discussion interesting. – Zev Chonoles Feb 22 '17 at 01:37
@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. – feralin Feb 22 '17 at 01:50

What is $F / F_{fin}$ isomorphic to, where $F$ is the group of integer functions and $F_{fin}$ is the subgroup of finitely-supported functions?

0 Answers0