Tag Archives: algebra

Einstein-Penrose ‘s strong sum convention

the rank one tensors’ basis changes



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short exact sequence and center

Let us prove:

Let 1\to A\stackrel{f}\to B\stackrel{g}\to B/A\to 1 be a short exact sequence, if the center Z(B/A)=1  then Z(B)<A

Proof:  When x\in Z(B) then g(x)\in Z(B/A), so g(x)=1.

Therefore x\in\ker (g)={\rm im}(f)=A



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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

Kurosh theorem à la Ribes-Steinberg

The strategic diagram is


to plainly grasp the technique


Filed under math

producto semi-directo

producto semi-directo

diagram chasing the wreath

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2014/05/03 · 14:50

permutational wreath product

Having an action G\times R\to R between two groups means a map (g,r)\mapsto ^g\!r that comply

  • {^1}r=r
  • ^{xy}r=\ ^x(^yr)
  • ^x(rs)=\ ^xr ^xs

Then one can assemble a new operation on R\times G to construct the semidirect product R\rtimes G. The group obtained is by operating

(r,g)(s,h)=(r\ {^h}s,g\ h).

Let \Sigma be a set and A^{\Sigma} the set of all maps \Sigma\to A. If we have an action \Sigma\times G\to\Sigma then, we also can give action G\times A^{\Sigma}\to A^{\Sigma} via


Then we define

A\wr_{\Sigma}G=A^{\Sigma}\rtimes G

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

Ribes – Steinberg 2008

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Filed under algebra, free group, group theory, math, maths, what is math

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