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)<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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otro lema estratégico


La siguiente lámina establece un lema vital para una demostración “más contemporánea” del Teorema de Sylow 1.

truccu

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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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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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transversal rewriting solution by semidirect product of certain coset maps


schreierx03este proceso se generaliza

transversalsandCM

 

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2014/08/21 · 13:48

not all is watching soccer


indeed

direcproduCfut

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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

Ribes – Steinberg 2008

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