Sylow theorems

Lagrange's theorem says a subgroup's order divides the group's order, but it does not promise a subgroup of every divisor. The Sylow theorems supply the strongest partial converse known: for the highest power of each prime dividing |G|, a subgroup of exactly that size always exists, and such subgroups are remarkably well-behaved. They are the single most powerful elementary tool for analyzing a finite group from its order alone.

Let |G| = p^k · m with p prime and p not dividing m. A Sylow p-subgroup is a subgroup of order p^k. The three theorems state: (1) Existence — Sylow p-subgroups exist, and in fact every p-subgroup is contained in one. (2) Conjugacy — any two Sylow p-subgroups are conjugate in G, so in particular all are isomorphic. (3) Counting — the number n_p of Sylow p-subgroups satisfies n_p ≡ 1 (mod p) and n_p divides m; equivalently n_p = [G : N_G(P)] for any Sylow p-subgroup P.

The counting constraints are sharp enough to pin groups down completely in small cases: if n_p = 1 the Sylow p-subgroup is normal, and many orders force exactly this. This is the standard route to proving a group of a given order cannot be simple, and hence to classifying small groups. The proofs themselves are elegant applications of group actions, the class equation, and the orbit-stabilizer theorem.