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
Ribes – Steinberg 2008
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different companion matrices
the double coset counting formula is a relation inter double cosets , where and subgroups in . This is:
The proof is easy.
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 $q_i$ 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 … This implies and so for all the posible , i.e, is normal.
Filed under maths, math, algebra, math analysis, fiber bundle, what is math, what is mathematics, mathematics, category theory, categoría, group theory
estas son las matemáticas antes llamadas “puras”
o no? :D