Category Archives: word algebra

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

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

 

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made in México


maybe, for the presentation \langle a,b,c\mid a^2=1, b^2=1 , c^2=1\rangle, is this the its Cayley’s graph?Nsub3CayleyGcC

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Filed under algebra, group theory, low dimensional topology, math, topology, word algebra

a poem, a strategic lemma


Lemma

\cdot  p\in{\Bbb{P}}

\cdot |G|=p^mr , p\nmid r

\cdot \forall H<G

\cdot \forall Q\in {\rm p\!-\!SS}_G

\Longrightarrow

\bullet \exists g\in G such that H\cap gQg^{-1}\in {\rm p\!-\!SS}_H.

Proof:

By employing the Double Coset Counting Formula we have |G|=\sum_a\frac{|H|\ |Q|}{|H\cap aQa^{-1}|}, and since p\nmid [G:Q] then \exists b\in\{a\} such that p\nmid [H:H\cap bQb^{-1}].

But H\cap bQb^{-1}<bQb^{-1} so |H\cap bQb^{-1}|=p^l\ ,\ \exists l\in{\Bbb{N}}, hence, having p\nmid\frac{|H|}{p^l}, this implies that |H|=p^l\alpha, where p\nmid\alpha\ ,\ \exists \alpha\in{\Bbb{N}}.

Then, for H\cap bQb^{-1}<H with |H\cap bQb^{-1}|=p^l we deduce it is {\rm p\!-\!SS}_H

\Box

DCCF01

double coset counting formula

 

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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

transversal rewriting solution by semidirect product of certain coset maps


schreierx03este proceso se generaliza

transversalsandCM

 

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2014/08/21 · 13:48