Prove that for $n>4$ every permutation in $S_n$ can be written as the product of 4-cycles.
Consider $\alpha=(a_1 a_2 a_3)(a_4 a_5)$ is the product of two disjoint cycles and $\alpha\in S_5$. We can write (according to a previous theorem) the following number of cycles $\alpha=(a_1 a_3)(a_1a_2)(a_4 a_5)$, which is less than 4.
1) Is $\alpha=(a_1 a_3)(a_1a_2)(a_4 a_5)$ the biggest number of cycles that can be written on this particular case? I have only learnt this technique.
2) How do I maximize or minimize the number of cycles for a given permutation?