Several days ago, Professor Martin Bridson gave a very nice talk in my department. Some questions concerning his talk came into my brain Since I am neither a model theorist nor a algebraist, I am not sure whether they fit here or not.

**Question 1. Given a finite group $Q$, is there a sentence $\varphi$ in the group language so that for any group $G$, $G\models \varphi$ if and only if there is a surjective homomorphism from $G$ to $Q$?**

Now let $G$ be a residually finite group and the corresponded inverse system $A=\{G/N\mid G/N \mbox{ is finite}\}$. Define $\hat{\Gamma}$ to be the inverse limit of $A$. Martin only focused on countable group and he mentioned that there is a quite important conjecture which says that for any two countable residually finite groups $G_0$ and $G_1$, $\hat{G_0}\cong\hat{G_1}\implies G_0\cong G_1$.Now I have the following question.

**Question 2. Is it true that for arbitrary residually finite groups $G_0$ and $G_1$ (they do not need to be countable), $\hat{G_0}\cong\hat{G_1}\implies G_0\equiv G_1$, where $\equiv$ means elementary equivalence relation?**

I am also interested the relationship between $G$ and $\hat{G}$. For example, $G\equiv_1 \hat{G}?$, where $\equiv_1$ means that they satisfy exactly same $\Sigma_1$-formulas?

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