Most of the proofs of Sylow theorems involves groups actions in some way as below:
There is a thread for them here, too: Proofs of Sylow theorems.
However, I would like to see proofs of these 3 theorems without using group actions. Somehow, they do not seem natural to me. If there is a reason for defining group actions, using orbits & stabilizers that I do not know, I would like to know it.