ultimate Sylow theorems version
An optimal guide for the proof of the Sylow’s theorem is offered. You must be acquainted with these concepts:
- group-action, with
- and conjugation-class-equation
Now let be a prime integral number. Then two aplications of that machinery are:
- if with , then the center
- if , then is abelian
Lemma: if in any group then .
The following sequence gonna take you to the mastery of a celebrated results on the theory of finite groups:
1. Cauchy (abelian) Theorem:
- Let be a prime integral number
- let be a finite abelian group
Then : .
The proof is inductive: we are going to admit that the proposition is true abelian such that
Let and . Then consider two cases: either or .
1-: . . .
2-: . . where . . such that . . . . where . . . . . .
2. Sylow I
- finite abelian :
The proof is based on double induction that is, we are going to accept the validity of proposition for any group such that and on .
[(n=1)]: Let where . Consider the two cases: either or .
1-: since is abelian then such that . So, with .
2-: since then such that . Then are coprime. . . . :
[(n=2)]: Let be, where . Consider again the two cases: either or .
1-: such that . , so with . With the natural epimorphism find . is also an epimorphism with . By the first fundamental homomorphisms theorem then .
2-: this implies . Since then such that , so they are coprime. . . . :
Now the induction on is clear
Definition of a -Sylow subgroup
3. Sylow II:
4. Sylow III: