Let D8 be the group of symmetries of the square. (a) Show that D8 can be...

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) reflections is a generating set of D8?

Note: What is mainly required here is patience. The first important step is to set up your notation in a clear way, so that you (and your reader) can see what you are doing. You might find it useful to write out the whole group table for D8, which is a useful exercise anyway. Then for part (a), choose one of the four reflections, think about how it composes with the rotation through 90◦, and how you can use this to obtain the remaining reflections. Try to explain why your argument would work for any of the four reflections. For parts (b) and (c), think about the geometry of the different pairs of reflections that you could choose. The composition of two reflections is always a rotation, but how does the angle of rotation depend on the two reflections that you choose?

Solutions

Expert Solution

part b) implies that D8 can be generated using 2 reflections.

part c) implies that any arbitrary of a pair will not always generate D8.

Colby Messinger answered 2 years ago

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