Sylow

Sylow refers to results in finite group theory known as the Sylow theorems, named after Ludwig Sylow who published them in 1872. The theorems describe the existence, number, and conjugacy properties of p-subgroups for finite groups.

Existence: Let G be a finite group and p a prime dividing |G|. Writing |G| = p^n m with p ∤ m, there exists a subgroup of G of order p^n. Such subgroups are called Sylow p-subgroups, and they are maximal p-subgroups of G.

Conjugacy: Any two Sylow p-subgroups of G are conjugate to each other, i.e., if P and Q are Sylow p-subgroups, there exists g in G with gPg^{-1} = Q.

Number: The number n_p of Sylow p-subgroups satisfies two conditions: n_p ≡ 1 mod p and n_p divides m = |G|/p^n. In particular, n_p > 1 unless the Sylow p-subgroup is normal.

Consequences and applications: The theorems constrain the possible structure of G, aid in proving normality of subgroups, and are instrumental in the study and classification of finite groups, including simple groups. They also yield counting arguments used to deduce existence of normal subgroups in various group orders.

History: The results are named after Sylow; he established the core statements for finite groups, with later refinements and alternative proofs appearing in algebra texts.