Sylow Theorems - Sylow Theorems

Sylow Theorems

Collections of subgroups that are each maximal in one sense or another are common in group theory. The surprising result here is that in the case of Syl_p(G), all members are actually isomorphic to each other and have the largest possible order: if |G| = pnm with n > 0 where p does not divide m, then any Sylow p-subgroup P has order |P| = pn. That is, P is a p-group and gcd(|G : P|, p) = 1. These properties can be exploited to further analyze the structure of G.

The following theorems were first proposed and proven by Ludwig Sylow in 1872, and published in Mathematische Annalen.

Theorem 1: For any prime factor p with multiplicity n of the order of a finite group G, there exists a Sylow p-subgroup of G, of order pn.

The following weaker version of theorem 1 was first proved by Cauchy.

Corollary: Given a finite group G and a prime number p dividing the order of G, then there exists an element of order p in G .

Theorem 2: Given a finite group G and a prime number p, all Sylow p-subgroups of G are conjugate to each other, i.e. if H and K are Sylow p-subgroups of G, then there exists an element g in G with g−1Hg = K.

Theorem 3: Let p be a prime factor with multiplicity n of the order of a finite group G, so that the order of G can be written as pnm, where n > 0 and p does not divide m. Let n_p be the number of Sylow p-subgroups of G. Then the following hold:

• n_p divides m, which is the index of the Sylow p-subgroup in G.
• n_p ≡ 1 mod p.
• n_p = |G : N_G(P)|, where P is any Sylow p-subgroup of G and N_G denotes the normalizer.