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**Pinkk** Let $\displaystyle N$ denote the group of isometries of a line $\displaystyle \mathbb{R}^{1}$. Classify discrete subgroups of $\displaystyle N$, identifying those that differ in the choice of origin and unit length on the line.

So I know the isometries of the line are translations and reflections, but I don't quite understand what the question is asking for. Obviously the composition of two distinct reflections in the line is a translation in the line, so I'm drawing a blank as to what the discrete subgroups are and what the "identifying..." part of the question is asking for. Any help would be appreciated, thanks.