# The group of sequences in $G^{\mathbb{N}}$ that converge to $(0,0,0,\dots)$

Let $$G$$ be a discrete abelian group and $$G^{\mathbb{N}}$$ be the direct product (with the product topology), which consists of sequences $$(a_1,a_2,a_3,\dots)$$ of elements of $$G$$.

Let $$G^{\infty}$$ denote the group of sequences with values in $$G^{\mathbb{N}}$$, which converge to the identity $$(0,0,0,\dots)\in G^{\mathbb{N}}$$. Hence, $$G^{\infty}$$ is naturally a subgroup of $$(G^{\mathbb{N}})^{\mathbb{N}}$$.

Is it possible to characterize the isomorphism type of $$G^{\infty}$$ in terms of the group $$G$$ or, perhaps, in terms of the structural characterization of infinite abelian groups, à la Fuchs, given a structural characterization of $$G$$?

I am curious about the general problem but am most interested in the case where $$G$$ is finitely generated. Also, I am aware of several results that help identify the isomorphism type of various subgroups of the Baer-Specker group $$\mathbb{Z}^{\mathbb{N}}$$ but it's not so clear to me how this applies or if it informs this more general situation.

• I think your meaning is clear, but, just to make sure: a "sequence in $G^{\mathbb N}$" could mean a sequence that is an element of $G^{\mathbb N}$ or that has values in $G^{\mathbb N}$. You mean the latter, right? Feb 6, 2021 at 17:32
• Correct. I do believe "converges to $(0,0,0,\dots)$" and $G^{\infty}\leq (G^{\mathbb{N}})^{\mathbb{N}}$ make this pretty clear but I guess I can try to clarify more. Feb 6, 2021 at 18:40
• A more robust and standard notation: for an arbitrary set $G^X$ is the group of maps $X\to G$, and $G^{(X)}$ is its subgroup of finitely supported functions (= which converge to 0 as sequences, when $G$ is discrete and $X=\mathbf{N}$).
– YCor
Feb 6, 2021 at 19:29
• @YCor I imagine $G^{(\mathbb{N})}$ is relevant to the problem but I'm having a hard time seeing how it could be used to rephrase the question to be more clear than it is now. Feb 6, 2021 at 20:10
• Don't take so hard time, I didn't suggest this.
– YCor
Feb 7, 2021 at 0:24