Let D8 be the group of symmetries of the square.
(a) Show that D8 can be generated by the rotation through 90?and any one of the four reflections.
(b) Show that D8 can be generated by two reflections.
(c) Is it true that any choice of a pair of (distinct) reflectionsis a generating set of D8?
Note: What is mainly required here is patience. The firstimportant step is to set up your notation in a clear way, so thatyou (and your reader) can see what you are doing. You might find ituseful to write out the whole group table for D8, which is a usefulexercise anyway. Then for part (a), choose one of the fourreflections, think about how it composes with the rotation through90?, and how you can use this to obtain the remaining reflections.Try to explain why your argument would work for any of the fourreflections. For parts (b) and (c), think about the geometry of thedifferent pairs of reflections that you could choose. Thecomposition of two reflections is always a rotation, but how doesthe angle of rotation depend on the two reflections that youchoose?
