Sylow theorems

In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow (1872) that give detailed information about the number of subgroups of fixed order that a given finite group contains. The Sylow theorems form a fundamental part of finite group theory and have very important applications in the classification of finite simple groups.Theorem 1: A finite group G whose order |G| is divisible by a prime power pk has a subgroup of order pk.Lemma: Let G be a finite p-group, let Ω be a finite set, let ΩG be the set generated by the action of G on all the elements of Ω, and let Ω0 denote the set of points of ΩG that are fixed under the action of G. Then |ΩG| ≡ |Ω0| (mod p). Theorem 2: If H is a p-subgroup of G and P is a Sylow p-subgroup of G, then there exists an element g in G such that g−1Hg ≤ P. In particular, all Sylow p-subgroups of G are conjugate to each other (and therefore isomorphic), that is, 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 q denote the order of any Sylow p-subgroup P of a finite group G. Then np = |G : NG(P)|, np ∣ {displaystyle mid } |G|/q and np ≡ 1 (mod p). In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow (1872) that give detailed information about the number of subgroups of fixed order that a given finite group contains. The Sylow theorems form a fundamental part of finite group theory and have very important applications in the classification of finite simple groups. For a prime number p, a Sylow p-subgroup (sometimes p-Sylow subgroup) of a group G is a maximal p-subgroup of G, i.e., a subgroup of G that is a p-group (so that the order of every group element is a power of p) that is not a proper subgroup of any other p-subgroup of G. The set of all Sylow p-subgroups for a given prime p is sometimes written Sylp(G). The Sylow theorems assert a partial converse to Lagrange's theorem. Lagrange's theorem states that for any finite group G the order (number of elements) of every subgroup of G divides the order of G. The Sylow theorems state that for every prime factor p of the order of a finite group G, there exists a Sylow p-subgroup of G of order pn, the highest power of p that divides the order of G. Moreover, every subgroup of order pn is a Sylow p-subgroup of G, and the Sylow p-subgroups of a group (for a given prime p) are conjugate to each other. Furthermore, the number of Sylow p-subgroups of a group for a given prime p is congruent to 1 mod p. 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 Sylp(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 every 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 every 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 Augustin-Louis Cauchy, and is known as Cauchy's theorem. Corollary: Given a finite group G and a prime number p dividing the order of G, then there exists an element (and hence a subgroup) 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.