Sylow’s theorems . In abstract algebra , specifically group theory , Sylow theorems are named in honor of the mathematician Norwegian Ludwig Sylow that provide detailed information on the number of subgroups of order set contained in a finite group given. Sylow’s theorems are a fundamental part of finite group theory and have very important applications in the classification of simple finite groups .
For a prime number p , a Sylow p -subgroup of a group G is a maximal p -subgroup of G , that is, a subgroup whose order is a power of p and which is not strictly contained in another p -group. That is, it is a group of order p k that is not contained in any subgroup of order p r where k < r . The set of all Sylow subgroups in a group G is usually denoted as Syl p ( G ).
Summary
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- 1 Sylow and Lagrange
- 2 Sylow’s theorems
- 1 Consequences
- 2 Sylow theorems for infinite groups
- 3 Sources
Sylow and Lagrange
Sylow theorems they constitute the partial reciprocal Lagrange ‘s theorem which states that for every finite group G , the order of any subgroup must divide the order of G . Conversely, for any prime factor p of the order of a finite group G , there will be a p -Sylow subgroup of order p n where n is precisely the multiplicity of the prime factor p in the order of G and any subgroup with the same order it will also be a p- subgroup of Sylow.
All Sylow subgroups of a fixed group and a given cousin are conjugated to each other. Finally, Sylow’s last theorem establishes a condition on the possible number of p- sub-groups of Sylow, indicating that this number will be congruent to 1 module p .
Sylow’s theorems
In group theory it is common to find collections of subgroups that are maximal in some way or another. The relevant result here is that in the case of Syl p ( G ), all its elements are isomorphic to each other and have the highest possible order: if | G | = p n m with n > 0 where p does not divide m , so every p- subgroup of Sylow P has order | P | = p n . That is, P is a p -group and the gcf (| G : P |,p ) = 1. These properties can be used to further analyze the structure of G .
The following theorems were originally stated and proved by Ludwig Sylow in 1872, publishing them in Mathematische Annalen.
Sylow’s first theorem . For any prime factor p with multiplicity n in the order (m = p n t) of the finite group G , then there exists a subgroup of G , with order p j for all j such that 1 ≤j ≤ n <ref> Zaldívar: Introduction to group theory . The following is a weaker version first demonstrated by Augustin Louis Cauchy.
Cauchy Given a finite group G and a prime number p that divides the order of G , there is an element of order p in G .
Sylow’s Second Theorem Given a finite group G , and a prime number p that divides the order of G , then all p- subgroups of Sylow are conjugated to each other. It is, if H and K are p -subgrupos Sylow then there is an element g in G such that g -1 Hg = K .
Sylow’s Third Theorem Let p be a prime factor with multiplicity n in the order of the finite group G , so that the order of G can be written as p n m where n > 0 and p does not divide m . Is n p the number of p -subgrupos Sylow of G . Then it is fulfilled;
- n pdivides m , which is the index of the p -subgrupos Sylow of G .
- n p≡ 1 mod p .
- n p= <math> | G: N_G (P) | </math>, where P is any p -Sylow subgroup of G and N G denotes the normalizer .
Consequences
Sylow’s theorems imply that for a prime p , every p -Sylow subgroup has the same order p n . Conversely, any subgroup having order p n will necessarily be a p- subgroup of Sylow and isomorphic to the other p- subgroups of Sylow. Due to the maximality condition, if H is a p -subgroup of G then H is a subgroup of a p -subgroup of Sylow. An important consequence of the third theorem is that the condition n p= 1 is equivalent to saying that in this case, the only Sylow p-subgroup is a normal subgroup (there are groups that have normal subgroups but do not have normal Sylow subgroups, with S 4 being an example of this).
Sylow theorems for infinite groups
There is an analog to Sylow’s theorem for infinite groups. We define a Sylow p -subgroup in an infinite group as a p -subgroup (that is, a subgroup where the order of every element is a power of p ) maximal with respect to the inclusion among the set of all p -subgroups. The existence of such subgroups is guaranteed by Zorn’s motto . Template: Theorem
