§ Counting necklackes with unique elements
Count number of ways to form a necklace with
- Method 1: This is equivalent to counting modulo the subgroup generated by . That subgroup has size . So the size is .
- Method 2: A cycle is an equivalence class of elements along with all of its cyclic shifts ( , , , ). We are to count the number of equivalence classes. First pick a canonical element of each equivalence class of the form .