álgebra

this part of my web-WP-place to interact algebraicaly. I mean, to talk about today’s algebra:

• elementary?
• estructures:  groups,  rings,  fields, vector spaces, modules, Lie groups and algebras, universal algebra
• categories:
• functors or arrows inter categories
• applications in analysis, geometry, topology:
• applications elsewhere:
• notices:
• events:

• A CATEGORY IN MATH IS A FASHION TO ESTRUTURE MATH THEMSELVES INTO A COLLECTION OF SETS CALLED THE OBJECTS OF THE CATEGORY AND A COLLECTION OF MORPHISMS (or ARROWS)  WHICH ARE FUNCTIONS AMONG THE OBJECTS
• We already had studied the

• abstraction lemma one:

Given a partition of a set there is an equivalence relation  defined on such that the equivalence classes coincide with the components of the partition. That is . Conversely, an equivalence relation in a set determines a partition of  it.

• which is a tool to say something about a set if it is this to big to handle.
• .
• .
• .
• now, a categorification in the mother category of set and functions interacting with  a equivalence relation is:
•  .
• consider the math-phenomena  of relating sets by functions and anyone may ask: how one has to splitt a domain into disjoint subsets for a function to be locally constant?
• .
• abstraction lemma two:

Given a epijective-map then can be factored as where is the projection defined as that is surjective, and defined as that is bijective

proof:

• first, one has to construct a binary relation in which is an equivalence relation:
• if
• RE: since obviously for each 
• SI:  implies . Then , so
• TR: if and then and  . So since
• .
• the equivalence classes are . That is, subsets where the function is constant: if then
• .
• So it makes sense to speak about the partition and the arrow via
• .
• note that this assignation is surjective since then such that , so
• .
• now, it is possible to construct a map simply by
• .
• but still we have to check that this is indeed a function
• .
• we must convince ourselves that this way of relate it does not depends in the form we choose to represent an equivalence class, for if then and then . Hence
• .
• now let’s proof that is bijective:
• .
• IY: let be two equivalence classes such that , then , so and hence
• .
• SY: if is any element of by surjectivity of we can find such that , so applying , we get a class with
• .
• 

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(01.nov.2011)

Después de  PRELIMINARES: GRUPOS

1. Definición, orden de elemento, producto de subconjuntos, conmutadores, presentaciones,…
2. Subgrupos y subgrupos normales. Clases laterales. Teorema de Lagrange.
3. Grupo cociente.
4. Morfismos de grupos. Pequeño teorema de Fermat.

TÓPICOS de GRUPOS

1. Grupo simétrico. Teorema de Cayley. es simple.
2. Acciones de grupos en conjuntos. Ecuación de Clase. Orbitas e Isotropía.
3. Teoremas de Cauchy y Sylow.
4. Resoluciones. Teorema fundamental de grupos abelianos.
5. Schreier systems. Teorema de Schreier-Nielsen.
6. Productos libres. El ejemplo
7. HNN-extensiones y