La siguiente lámina establece un lema vital para una demostración “más contemporánea” del Teorema de Sylow 1.
Tag Archives: Sylow’s theorems
such that .
By employing the Double Coset Counting Formula we have , and since then such that .
But so , hence, having , this implies that , where .
Then, for with we deduce it is
the double coset counting formula is a relation inter double cosets , where and subgroups in . This is:
One is to be bounded to the study of the natural map . And it uses the second abstraction lemma.
The formula allows you to see the kinds of subgroups of arbitrary versus a of , for the set of the – Sylow subgroups.
Or, you can see that through the action via you can get:
- which comply the equi-partition
- , so , for some
then you can deduce:
Now, let us use those ideas to prove the next statement:
Let be a finite group, with cardinal , where each are primes with and positive integers.
Let be a subgroup of of index .
Then, is normal.
By employing in the double coset partition, one get the decomposition:
So by the double coset counting formula you arrive to:
From this, we get .
But as well so
Then . So for each .
This implies and so for all the posible , hence, is normal.
We are going to reconstruct the salient of a part of the classical theorem:
Each p-S of is contained into a p-SS of
Here the reference frame:
1. p-SS divides :
The set p-SS must be considered as an orbit of the action p-SS p-SS via . Since from sylow II we know that each two are conjugated the there is only one orbit
giving us a trivial orbital partition. Then p-SS=, i.e.
because for the isotropy group is .
2. Observe also that #p-SS:
since and , as far as
which implies that p-SS, then p-SS.
Proof of Theorem:
Considering the conjugation sub-action
we get a orbit decomposition p-SS = with the corresponding class equation
#p-SS = ,
but asumming that then
#p-SS = .
Now since #p-SS then there is some for which , so
With that, it is easy these: and is normal in .
Also given by is an epimorphism, so by the fundamental theorem of group-morphisms we have
Observing that and then
but is maximal, then . Hence and