Observation: Two disjoint permutations are commutative. For e.g $\ (1357)\in S_8$ and $\ (2468)\in S_8$. Formulate and prove(if possible) a Theorem generalizing your observation about commutative permutations.
Attempt of a solution:
Theorem: Two disjoint permutations commute
Proof: I am unable to prove the theorem. I can only show examples. Can anyone prove this theorem without examples?
On
Hint: Look at some $a\in[n]$ and denote the permutation as $$\pi=c_{1}c_{2}$$ where $c_{i}$ are two disjoint circles.
Consider the following cases: $$c_{1}(a)\neq a,c_{2}(a)\neq a,c_{1}(a)=c_{2}(a)=a$$
Do you see why the cases are disjoint and it must be the case that exactly one of them holds ?
Can you continue from here ?
You can write the permutations as matrices ("acting" on the vector (1 2...8)) , and show that the two commute when multiplied; not too hard, since you will have blocks of identity submatrices