Count number of ways to form a necklace with $\{1, 2, \dots, n\}$

• Method 1: This is equivalent to counting $|S_5|$ modulo the subgroup generated by $(12\dots)$. That subgroup has size $5$. So the size is $S_5/5$.
• Method 2: A cycle is an equivalence class of elements $(a,b,c,d,e)$ along with all of its cyclic shifts ( $(b,c,d,e,a)$, $(c,d,e,a,b)$, $(d,e,a,b,c)$, $(e,a,b,c,d)$). We are to count the number of equivalence classes. First pick a canonical element of each equivalence class of the form $(1, p, q, r, s)$.