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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 $\widetilde{\Pi}G \cong H$?

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    $\widetilde{\Pi}G$ is the [*direct sum*](http://mathworld.wolfram.com/GroupDirectSum.html) of $\omega$ copies of $G$.2012-03-01
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    For a counterexample to your final question (buried at the end of my answer), take $G=\mathbb{Z}^2\oplus (\mathbb{Z}/2\mathbb{Z})$ and $H=\mathbb{Z}\oplus(\mathbb{Z}/2\mathbb{Z})$. No finite power of $G$ is isomorphic to a finite power of $H$, nor are $G$ or $H$ isomorphic to the other's "reduced power" (restricted direct power), but $\displaystyle\widetilde{\prod}G \cong \widetilde{\prod}H$.2012-03-02

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