Let . Define the group
Prove that where is the dihedral group.
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 .
which is precisely the dihedral group with elements.