The importance of Sylow’s theorem in group theory is so immense that it can be hard to fully communicate, until after you’ve seen its varied applications!
This is maybe not the big one; maybe it doesn’t quite earn the title of the “Fundamental Theorem of Group Theory” (I’ll announce that one when it comes.).
But it certainly is a big one.
Just to not prolong the agony, I’ll tell you the statement of Sylow’s theorem as quickly as possible. The proofs and applications come toward the end.
I don’t love the ways that the book proves the Sylow theorems. Not because the book isn’t right and great — it is both.
But I think other, better proofs have become more widely known since the time of its publication. So I’ll prefer to use my favorite proofs of these theorems.
My favorite proofs require first proving a few antecedent facts. However, those antecedents are themselves interesting, besides also allowing us to prove Sylow’s theorem. So the juice is worth the squeeze!
In particular we will need the p-group fixed point theorem.