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:
- 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.
Filed under algebra, categoría, category theory, fiber bundle, group theory, math, math analysis, mathematics, maths, what is math, what is mathematics
estas son las matemáticas antes llamadas “puras”
o no? :D
are the common algebraic-techniques territory for today vector algebra and differential geometry.
This means that ancient vectorcalculus that turns into differential forms nowadays, is “super-oversimplified” into a mathematical language to phrase some modern geometricalalgebrotopologicalanalitic-maths.
Real, ‘cuz first you gotta get the ideas over the field .
Filed under algebra, calculus on manifolds, category theory, differential geometry, geometry, math analysis, mathematics, multilinear algebra, topology, what is math, what is mathematics
sabiendo que y que tanto como , entonces
Ahora para un covector arbitrario se tiene:
para todo .
Es decir el representante del covector –à la Riesz- es:
esto es una combinación lineal en la base recíproca de , which represent the basic covectors