Category Archives: math

my profession

not all is watching soccer


indeed

direcproduCfut

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Filed under group theory, math

Kurosh theorem à la Ribes-Steinberg


The strategic diagram is

Image

to plainly grasp the technique

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producto semi-directo


producto semi-directo

diagram chasing the wreath

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2014/05/03 · 14:50

wreath functoriality


wreath functoriality

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2014/04/03 · 21:20

heptapenti


heptapenti

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2014/03/12 · 18:55

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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Filed under algebra, free group, group theory, math, maths, what is math

2013 in review


The WordPress.com stats helper monkeys prepared a 2013 annual report for this blog.

Here’s an excerpt:

A New York City subway train holds 1,200 people. This blog was viewed about 5,200 times in 2013. If it were a NYC subway train, it would take about 4 trips to carry that many people.

Click here to see the complete report.

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