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.

$latex text{ }$


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:


para empezar, y preguntar más si quieren,


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