a group

SL_2(Z_3)

Group SL_2(\Bbb Z_3):

a=\left(\begin{array}{cc}0&1\\2&0\end{array}\right) , b=\left(\begin{array}{cc}0&1\\2&1\end{array}\right) , ba^3b=\left(\begin{array}{cc}0&1\\2&2\end{array}\right)

a^3=\left(\begin{array}{cc}0&2\\1&0\end{array}\right) , bab=\left(\begin{array}{cc}0&2\\ 1&1\end{array}\right) , a^2b=\left(\begin{array}{cc}0&2\\1&2\end{array}\right)

e=\left(\begin{array}{cc}1&0\\ 0&1\end{array}\right) , a^2ba=\left(\begin{array}{cc}1&0\\ 1&1\end{array}\right) , (ba)^2=\left(\begin{array}{cc}1&0\\2&1\end{array}\right)

(ab)^2=\left(\begin{array}{cc}1&1\\0&1\end{array}\right) , b^2ab=\left(\begin{array}{cc}1&1\\ 1&2\end{array}\right) , a^3ba=\left(\begin{array}{cc}1&1\\2&0\end{array}\right)

a^3b=\left(\begin{array}{cc}1&2\\0&1\end{array}\right) , a^2b^2=\left(\begin{array}{cc}1&2 \\1&0\end{array}\right) , bab^2=\left(\begin{array}{cc}1&2\\2&2\end{array}\right)

a^2=\left(\begin{array}{cc}2&0\\ 0&2\end{array}\right) , ab^2=\left(\begin{array}{cc}2&0\\ 1&2\end{array}\right) , ba=\left(\begin{array}{cc}2&0\\ 2&2\end{array}\right)

ab=\left(\begin{array}{cc}2&1\\ 0&2\end{array}\right) , b^2aba=\left(\begin{array}{cc}2&1\\ 1&1\end{array}\right) , b^2=\left(\begin{array}{cc}2&1\\2&0\end{array}\right)

b^2a=\left(\begin{array}{cc}2&2\\0&2\end{array}\right) , aba=\left(\begin{array}{cc}2&2\\1&0\end{array}\right) , abab^2=\left(\begin{array}{cc}2&2\\2&1\end{array}\right)

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A Clever Proof of a Common Fact

Cf. means confer…

Abstract Nonsense

Point of Post: In this post we give a new proof that if $latex G$ is a finite group $latex H$ is a subgroup $latex G$ whose index is the smallest prime dividing $latex |G|$ then that subgroup is normal.

$latex text{ }$

Motivation

It is a commonly used theorem in finite group theory that if $latex G$ is a finite group and $latex Hleqslant G$ such that $latex left[G:Hright]=p$ is the smallest prime dividing $latex |G|$ then $latex Hunlhd G$. We have already seen a proof of this fact by considering the homomorphism $latex Gto S_p$ which is the induced map from $latex G$ acting on $latex G/H$ by left multiplication, and proving that $latex ker(Gto S_p)=H$. We now give an even shorter (and the just mentioned proof is already short) proof of this fact using double cosets.

$latex text{ }$

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245B, Notes 10: Compactness in topological spaces

compact topology a la Terrence Tao

What's new

One of the most useful concepts for analysis that arise from topology and metric spaces is the concept of compactness; recall that a space $latex {X}&fg=000000$ is compact if every open cover of $latex {X}&fg=000000$ has a finite subcover, or equivalently if any collection of closed sets with the finite intersection property (i.e. every finite subcollection of these sets has non-empty intersection) has non-empty intersection. In these notes, we explore how compactness interacts with other key topological concepts: the Hausdorff property, bases and sub-bases, product spaces, and equicontinuity, in particular establishing the useful Tychonoff and Arzelá-Ascoli theorems that give criteria for compactness (or precompactness).

Exercise 1 (Basic properties of compact sets)

  • Show that any finite set is compact.
  • Show that any finite union of compact subsets of a topological space is still compact.
  • Show that any image of a compact space under a continuous…

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definición en diagrama

¿Cuál es la definición de un grupo libre?

me preguntaban…

y yo les digo (yo propongo) que deberían de considera un diagrama de conjuntos y flechas así:

FGdiagrmdefA

para empezar, y preguntar más si quieren,

continuo,

les digo que se pueden consultar cantidades tremendas de literatura, Schreier en 1927 resolvió el teorema fundamental:

Si F es un grupo libre y U<F entonces U también es libre

la respuesta para este teorema utiliza  este otro diagrama

FGdiagdefiA