Let be the usual presentation of the dihedral group of order and let be a positive integer dividing .
- Prove that is normal in .
- Prove that .
- By a previous exercise it suffices to note that and .
- Define a mapping by and . Note the following.
- Note that . Now if and only if . The least integer such that is thus .
- , so that .
- Suppose by way of contradiction that ; then we have for some integer , a contradiction.
By a lemma to a previous exercise, induces an injective group homomorphism . Moreover, is surjective because .