Let $G$ be a countable group. Consider the subgroup $\widetilde{\Pi}G = \left\{(g_n) \in \prod\nolimits_{\mathbb{N}} G\,:\,g_n \neq e\text{ only for finitely many }n\right\}$ of the countable power of $G$ with itself. Then $\tilde{\Pi}G$ is a countable group which one might call the reduced power of $G$. This leads me to the first question:
First question: is there an official name for $\widetilde{\Pi}G$?
For some bizarre reasons, I came to be interested in groups $G$ that are isomorphic to $\widetilde{\Pi}G$ and lacking any information about the name of this construction it is hard for me to start trying to find information on them. Rather vaguely my question is:
What can be said about the properties of $\widetilde{\Pi}G$? What properties of $G$ are reflected in $\widetilde{\Pi}G$? Is there a place where groups of the form $\widetilde{\Pi}G$ are studied?
Two simple observations about this construction:
If $G \cong \widetilde{\Pi}G$ then $G$ cannot be finitely generated because every finite set $S \subset \widetilde{\Pi}G$ generates a subgroup having only finitely many nontrivial coordinates. This means that many basic group theoretic properties aren't preserved in a straighforward way by this construction. (This makes me a bit doubtful about the value of this construction.)
About the only mildly interesting positive result I could find is: $\widetilde{\Pi}G$ is amenable if and only if $G$ is amenable. If $G$ is amenable then $\widetilde{\Pi}G$ is the union of the amenable groups $G^n$, and if $\widetilde{\Pi}G$ is amenable, then so is its subgroup $G$.
Since this site seems to favor specific questions, here's one:
Suppose $\widetilde{\Pi}G \cong \widetilde{\Pi} H$. Is it true that either $G^n \cong H^m$ for some $n,m$ or $G \cong \widetilde{\Pi} H$ or $\widetilde{\Pi}G \cong H$?