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Here is some thougths on infinite finitely generated groups. Let $sq\colon G\to G$ be the square function mapping $g$ to $g^2$. Hence the set of elements of order at most $2$ is equal to $sq^{-1}(1)$. …

Let $G$ be a finitely generated residually finite group and $\hat G$ its profinite completion. Then for all $g\in \hat G$ we have $gGg^{-1}\leq \hat G$ is dense. Suppose that $H\leq \hat G$ is a clos …

Question Let $G$ be a countably infinite reduced abelian $p$-group. Is it always possible to write it has an infinite direct sums of non-trivial groups? If it is not true, how far is $G$ from being an …

This is a reference request for the following statement: Fact: Let $G$ be a reduced abelian $p$-group with an element of order $p^2$. Then $G$ contains an element of order $p$ and of finite height at …

Here are some usefull facts, and some historical details. Every $2d$-regular graph (without loops of degree 1) is isomorphic to a Schreier graph. The results for finite graphs is due to Gross. The res …

Preliminaries: in the following, $G$ will always be a finitely generated group and $\mathrm{Schr}(G,H;S)$ will be the Schreier graph corresponding to the symmetric generating set $S^{\pm1}$. Schreier …