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Notice that isometry group of euclidean space $\Bbb{R}^n$ is displayed by $E(n)$.

I would like know that why any discrete subgroup $G$ of $E(n)$ ( i.e subspace topology (from $E(n)$) on $G$ is discrete) has locally finite orbits in $\Bbb{R}^n$ and vice versa.

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A subgroup $G$ of $E(n)$ is called discrete if for each point $x\in \mathbb{R}^n$ the family $\{g.x\mid g\in G\}$ is locally finite. Since this family contains each point with multiplicity equal to the order of the stabiliser $G_x$ of $x$, the family is locally finite if and only if the orbit $Gx$ is discrete and the stabilizer $G_x$ is finite. This shows that the subgroup $G$ is discrete with respect to the subset topology. Hence the two usual definitions are equivalent. Furthermore $G$ is a discrete subgroup of $E(n)$ if and only if $G$ acts properly discontinuously on $\mathbb{R}^n$.

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