Let

denote the group of isometries of a line

. Classify discrete subgroups of

, 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.