Having an action between two groups means a map that comply
Then one can assemble a new operation on to construct the semidirect product . The group obtained is by operating
Let be a set and the set of all maps . If we have an action then, we also can give action via
Then we define
the so called permutational wreath product.
This ultra-algebraic construction allow to give a proof of two pillars theorems in group theory: Nielsen – Schreier and Kurosh.
The proof becomes functorial due the properties of this wreath product.
The following diagram is to be exploited
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Let be the rank two free group and be a subgroup.
Observe that , then .
Clearly , because it is not difficult to convince oneself that consists on words of even length and implies .
Technically, that is attending to the Schreier’s recipe, having as a set of transversals and being the free generators for .
Set and take , then we get
So according to Schreier’s language the set in our case, is
Hence are the free generator for .
Note that this three word are the first three length-two-words in the alphabetical order, start by and continuing to
different companion matrices
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:
- , and
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.