# Category Archives: group theory

## 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

## transversal rewriting solution by semidirect product of certain coset maps

2014/08/21 · 13:48

## not all is watching soccer

Filed under group theory, math

## 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