Let . Define the group

Prove that where is the dihedral group.

**Solution**

Using or equivalently we can write each element of in the form where . Using we may assume that . Using we may also assume that . It is easy to prove inductively that .

Let . We prove that . Obviously . Furthermore, since

If (such that ) then hence and therefore or . If then hence which is a contradiction since .

Therefore,

which is precisely the dihedral group with elements.