Category Archives: maths
Let us prove:
Let be a short exact sequence, if the center then
Proof: When then , so .
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
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.