Category Archives: mathematics

Einstein-Penrose ‘s strong sum convention

the rank one tensors’ basis changes



Filed under math, mathematics, multilinear algebra, word algebra

wedge product example

When bivectors are defined by


so, for two generic covectors

\theta=a\beta^1+b\beta^2+c\beta^3 and \phi=d\beta^1+e\beta^2+f\beta^3,

we have the bivector




Cf. this with the data \left(\begin{array}{c}a\\b\\c\end{array}\right) and \left(\begin{array}{c}d\\e\\f\end{array}\right) to construct the famous


So, nobody should be confused about the uses of the symbol \wedge dans le calcul vectoriel XD


Filed under algebra, cucei math, differential geometry, math analysis, mathematics, multilinear algebra, what is math, word algebra

the rank 3 free group is embeddable in the rank two free group

Let F=\langle x,y|\ \rangle be the rank two free group and U=\langle\{x^2,y^2,xy\}\rangle be a subgroup.
Observe that xy^{-1}=xy(y^2)^{-1}, then xy^{-1}\in U.

Clearly F=U\sqcup Ux, because it is not difficult to convince oneself that U consists on words of even length and xy^{-1}\in U implies Uy=Ux.

Technically, that is attending to the Schreier’s recipe, having \Sigma=\{1,x\} as a set of transversals and being S=\{x,y\} the free generators for F.

Set \Sigma S=\{x,\ y,\ x^2,\ xy\} and take \overline{\Sigma S}=\{1,x\}, then we get

\overline{\Sigma S}^{-1}=\{1,x^{-1}\}.

So according to Schreier’s language the set  \Sigma S\overline{\Sigma S}^{-1}=\{ gs\overline{gs}^{-1}|g\in\Sigma,s\in S\}, in our case, is

\{\ x\overline{x}^{-1}=1\ ,\ y\overline{y}^{-1}=yx^{-1}\ , \ x^2\overline{x^2}^{-1}=x^2\ ,\ xy\overline{xy}^{-1}=xy\ \}.

Hence \{\ xy^{-1}\ ,\ x^2\ ,\ xy\ \} are the free generator for U.

Note that this three word are the first three length-two-words in the alphabetical order, start by  1<x<x^{-1}<y<y^{-1} and continuing  to


. . .< yx<yx^{-1}<y^2<y^{-1}x<y^{-1}x^{-1}<y^{-2}



Filed under algebra, cucei math, free group, group theory, math, mathematics, what is math, what is mathematics

cf. Frobenius forms

cf. Frobenuis forms

different companion matrices


Filed under algebra, math, mathematics, what is math, what is mathematics

double coset counting formula

the double coset counting formula is a relation inter double cosets HaK, where a\in G and H,K subgroups in G. This is:

\#(HaK)=\frac{|H||K|}{|H\cap aKa^{-1}|}


\#(G/K)=\sum_a[H;H\cap aKa^{-1}]

The proof is easy.

One is to be bounded to the study of the natural map H\times K\stackrel{\phi_a}\to HaK. And it uses the second abstraction lemma.

The formula allows you to see the kinds of subgroups of arbitrary H versus K a p-SS of G, p-SS for the set of the p– Sylow subgroups.

Or, you can see that through the action H\times G/K\to G/K via h\cdot aK=haK you can get:

  • {\rm Orb}_H(aK)=\{haK\} which comply the equi-partition
  • HaK=aK\sqcup haK\sqcup...\sqcup h_taK, so \#(HaK)=m|K|, for some m\in \mathbb{N}
  • {\rm St}_H(aK)=H\cap aKa^{-1}

then you can deduce:

|G|=\sum_a\frac{|H||K|}{|H\cap aKa^{-1}|}

Now, let us use those ideas to prove the next statement:

Let G be a finite group, with cardinal |G|=q_1^{n_1}q_2^{n_2}\cdots q_t^{n_t}, where each q_i are primes with q_1<q_2<...<q_t and n_i positive integers.

Let H be a subgroup of |G| of index [G:H]=q_1.

Then, H is normal.


By employing K=H in the double coset partition, one get the decomposition:

G=HeH\sqcup Ha_1H\sqcup...\sqcup Ha_tH

So by the double coset counting formula you arrive to:

|G/H|=1+[H:H\cap a_1Ha_1^{-1}]+\cdots+[H:H\cap a_tHa_t^{-1}]


q_1=1+\frac{|H|}{|H\cap a_1Ha_1^{-1}|}+\cdots+\frac{|H|}{|H\cap a_tHa_t^{-1}|}

From this, we get \frac{|H|}{|H\cap a_iHa_i^{-1}|}<q_1.

But |G|=q_1|H| as well |H|=|H\cap a_iHa_i^{-1}|[H:H\cap a_iHa_i^{-1}] so

|G|=q_1|H\cap a_iHa_i^{-1}|[H:H\cap a_iHa_i^{-1}], i.e.

[H:H\cap a_iHa_i^{-1}] divides |G|

Then [H:H\cap a_iHa_i^{-1}]=1. So |H|=|H\cap a_iHa_i^{-1}| for each a_i.

This implies H=H\cap a_iHa_i^{-1} and so H=a_iHa_i^{-1} for all the posible a_i, hence, H is normal.



Filed under algebra, categoría, category theory, fiber bundle, group theory, math, math analysis, mathematics, maths, what is math, what is mathematics

real elementary multilinear algebra

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 \mathbb{R}.


Filed under algebra, calculus on manifolds, category theory, differential geometry, geometry, math analysis, mathematics, multilinear algebra, topology, what is math, what is mathematics

función con círculos críticos en el toroide

to see the example à la wiki of a function with circles as a critical sets of a  scalar field in a surface in three dimensional space:  follows…

para ver un ejemplo à la wiki de una función con cercos como uns insiemi o campo escalar en una área del trosième Raum: sigue…



actualización (15.Nov.2011):

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Filed under calculus on manifolds, differential geometry, low dimensional topology, mathematics, topology