Tag Archives: algebra

2015/12/08 · 17:50

Einstein-Penrose ‘s strong sum convention

the rank one tensors’ basis changes

abstractrelativity017

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Filed under math, mathematics, multilinear algebra, word algebra

Tagged as algebra, álgebra multilineal, cucei, linear algebra, multilinear algebra, tensors, Universidad de Guadalajara

2015/05/22 · 14:15

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

$\Box$

wedge product example

When bivectors are defined by

$\beta^i\wedge\beta^j:=\beta^i\otimes\beta^j-\beta^j\otimes\beta^i$ ,

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

$\theta\wedge\phi=(bf-ce)\beta^2\wedge\beta^3+(cd-af)\beta^3\wedge\beta^1+(ad-be)\beta^1\wedge\beta^2$ .

Otherwise,

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

$\left(\begin{array}{c}a\\b\\c\end{array}\right)\times\left(\begin{array}{c}d\\e\\f\end{array}\right)=\left(\begin{array}{c}bf-ce\\cd-af\\ad-be\end{array}\right)$

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

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

Tagged as algebra, analyse vectorielle, álgebra exterior, bivector, calcul vectoriel, calculus, covector, linear algebra, multilinear algebra, vector analysis, wedge product

2014/01/28 · 14:06

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

$gf(x)=f(xg)$

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

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:

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

and

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

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.