Let $G$ be a Hausdorff topological group. We say that a subgroup $S$ of $G$ is discrete if and only if the subspace topology (from $G$) on $S$ is discrete.
Note that isometry group of euclidean space $\mathbb{R}^n$ is displayed by $E(n)$(or affine isometries on $\mathbb{R}^n , \ O(n)\subset E(n)$) and $Sx:=\left\{s(x);\ \forall s\in S\subseteq E(n)\right\}$ is $S$-orbit of any $x\in\mathbb{R}^n$ where S is any subgroup of E(n).
For discrete subgroups S of isometry group of euclidean space $E(n)$ an equivalent condition is: intersection of the $S$-orbit of any $x\in\mathbb{R}^n$ has finite intersection with any compact set of $\mathbb{R}^n$.
Why is there a such the equivalent condition? How can I prove it?
The definition means that $S$ is discrete if for each point $x\in \mathbb{R}^n$ the family $Sx$ is locally finite, i.e., for each point there is a neighbourhood intersecting only finitely many subsets of this family. This is equivalent to the fact each compact set intersects only finitely many subsets of this family.