Abstract Algebra (4): Sylow Theorems — Dissecting Finite Groups

Tue, 07 Sep 2021 09:00:00 +0000

Lagrange’s theorem tells you the order of any subgroup must divide $$|G|$$ . That is a necessary condition, and it is famously not sufficient — the alternating group $$A_4$$ has order $$12$$ , but no subgroup of order $$6$$ . So the moment you start asking “given $$|G| = n$$ , what does $$G$$ actually look like?”, Lagrange leaves you holding an empty bag.

The Sylow theorems are what go inside that bag. They say: for every maximal prime power $$p^a$$ dividing $$|G|$$ , a subgroup of order $$p^a$$ exists, all such subgroups are conjugate, and their count $$n_p$$ is sharply constrained ( $n_p \equiv 1 \pmod p$ , $n_p \mid [G:P]$ ). Ludwig Sylow proved this in 1872, and 150 years later it is still the first thing you reach for when somebody hands you a finite group of unknown order and asks what it is.

Sylow-Theorems on Chen Kai Blog

Abstract Algebra (4): Sylow Theorems — Dissecting Finite Groups