When bivectors are defined by
so, for two generic covectors
we have the bivector
Cf. this with the data and to construct the famous
So, nobody should be confused about the uses of the symbol dans le calcul vectoriel XD
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
álgebra multilineal es como un cálculo vectorial dos o álgebra lineal tres
entonces para poder hacer cálculos en otras geometrías, inclusive muy diferentes a vamos viendo hacia donde tenemos que caminar: ver (un post previo con estas ideas en mente).
RE-ENGINEERING LINEAR ALGEBRA
RE-ENGINEERING VECTOR CALCULUS
the group is , and the picture:
Correctly predicts the next one. It is A164001 in the OEIS data-base. Dubbed “Spiral of triangles around a hexagon“. It has the generating function , Why would it be? :) . This another is A000931.
what is math? let us discuss:
|Baby Abstract Multilinear Algebra
|Baby Multilinear Algebra of Inner Product Spaces
|Algebraic Differential Geometry
- Parameterizations: curves and surfaces
- Tangent vectors, tangent space, tangent bundle
- Curves in and and on surfaces in
- Surfaces in
- all classical surfaces rendered
- tangent space change of basis
- vector fields and tensor fields
- Christoffel’s symbols (connection coefficients)
- Curvatures (Gaussian, Mean, Principals, Normal and Geodesic)
- Vector Fields, Covector Fields, Tensor Fields
- Integration: Gauss-Bonnet, Stokes
|Baby Manifolds (topological, differential, analytic, anti-analytic, aritmetic,…)
|Examples: Lie groups and Fiber bundles
Filed under algebra, calculus on manifolds, categoría, differential equations, differential geometry, fiber bundle, geometry, math analysis, maths, multilinear algebra, topology, what is math
in this post I am gonna again insist about the cruel confusion spawn by people who, I think, misunderstand certain strategic points in the grasping of tensors.
Vectors in are linear combinations of
, , … , ,
that is, vectors are expressions as , or employing the Einstein-Penrose convention: .
But, covectors are linear combinations of
, , … , ,
which are also dubbed “projectors” due its role as a linear maps .
So a general covector is a linear combination , using the Einstein-Penrose Convention again.
It is regrettable how many authors are merciless in the misuses of the rows and columns concepts for vectors and covectors respectively. It seems that sacrificing the difference by writing vectors as rows to gain space in the texts of linear algebra books is a big cost in the habits of understand well the matter.
This is perhaps a simple reason why it is so difficult to everybody to grasp the idea of tensors at its first steps. Well, not everybody … hehehe.
We say that duality rules ‘cuz
becomes a simple device to serve as a guide for a (apparently) complex calculations that arise when one consider tensors of higher rank.
Once recognized this hurdle, the constructions of a calculus à la Cartan, using the wedge product and the exterior derivative among tensor fields is an easy cake. These techniques will allow you to free of the ugly vector calculus that rules in each bachelor science schools nowadays.