  For a group $G$, we denote by $\stackrel{\leftrightarrow}{G}$ the coarse
space on $G$ endowed with the coarse structure with the base $\{\{ (x,y)\in
G\times G: y\in x^F \} : F \in [G]^{<\omega} \}$, $x^F = \{z^{-1} xz : z\in F
\}$. Our goal is to explore interplays between algebraic properties of $G$ and
asymptotic properties of $\stackrel{\leftrightarrow}{G}$. In particular, we
show that $asdim \ \stackrel{\leftrightarrow}{G} = 0$ if and only if $G / Z_G$
is locally finite, $Z_G$ is the center of $G$. For an infinite group $G$, the
coarse space of subgroups of $G$ is discrete if and only if $G$ is a Dedekind
group.
