Indicate schematically a Cayley digraph for D,, using a generating set consisting of a rotation through 2rr / n radians and a reflection (mirror image). See Exercise 44.

Data from Exercise 44.

In analogy with Examples 8.7 and 8.10, consider a regular plane n-gon for n ≥ 3. Each way that two copies of such an n-gon can be placed, with one covering the other, corresponds to a certain permutation of the vertices. The set of these permutations in a group, the nth dihedral group D_n• under permutation multiplication. Find the order of this group D_n. Argue geometrically that this group has a subgroup having just half as many elements as the whole group has.