# Category Archives: algebra

## Einstein-Penrose ‘s strong sum convention

the rank one tensors’ basis changes 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)

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

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

Filed under 3-manifold, algebra, group theory, word algebra

## otro lema estratégico

La siguiente lámina establece un lema vital para una demostración “más contemporánea” del Teorema de Sylow 1. Filed under algebra, cucei math, group theory, what is math

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? 1 Comment

## a poem, a strategic lemma

Lemma $\cdot$ $p\in{\Bbb{P}}$ $\cdot$ $|G|=p^mr$ , $p\nmid r$ $\cdot$ $\forall H $\cdot$ $\forall Q\in {\rm p\!-\!SS}_G$ $\Longrightarrow$ $\bullet$ $\exists g\in G$ 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} 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} with $|H\cap bQb^{-1}|=p^l$ we deduce it is ${\rm p\!-\!SS}_H$ $\Box$

Filed under algebra, group theory, math, word algebra

## 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