# 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: $\{\{\rm{OBJS}\},\{\rm{MORPHS}\}\}$
• 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
• abstraction lemma one:

Given a partition $\Pi$ of a set $S$ there is an equivalence relation $\sim$ defined on $S$ such that the equivalence classes coincide with the components of the partition. That is $\Pi=S/{\sim}$. 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 $f:S\to T$ then $f$ can be factored as $f=\beta\circ\rho$ where $\rho$ is the projection $S\to S/{\sim}$ defined as $\rho(s)=[s]$ that is surjective, and $\beta:\frac{S}{\sim}\to T$ defined as $\beta([x])=f(x)$ that is bijective

proof:

• first, one has to construct a binary relation in $S$ which is an equivalence relation:
• $x\sim y$ if $f(x)=f(y)$
• RE: $x\sim x$ since $f(x)=f(x)$ obviously for each  $x\in S$
• SI: $x\sim y$ implies $f(x)=f(y)$. Then $f(y)=f(x)$, so $y\sim x$
• TR: if $x\sim y$ and $y\sim z$ then $f(x)=f(y)$ and  $f(y)=f(z)$. So $x\sim z$ since $f(x)=f(z)$
• .
• the equivalence classes are $[a]=\{x\in S\mid f(a)=f(x)\}$. That is, subsets where the function is constant: if $y\in [a]$ then $f(y)=f(a)$
• .
• So it makes sense to speak about the partition $S/{\sim}$ and the arrow $\rho: S\to S/{\sim}$ via $\rho(s)=[s]$
• .
• note that this assignation is surjective since $C\in S/{\sim}$ then $\exists c\in S$ such that $[c]=C$, so $\rho(c)=[c]=C$
• .
• now, it is possible to construct a map $\beta:\frac{S}{\sim}\to T$ simply by $\beta([x])=f(x)$
• .
• but still we have to check that this $\beta$ 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 $[a]=[b]$ then $a\sim b$ and then $f(a)=f(b)$. Hence $\beta([a])=\beta([b])$
• .
• now let’s proof that $\beta$ is bijective:
• .
• IY: let $[a],[b]$ be two equivalence classes such that $\beta([a])=\beta([b])$, then $f(a)=f(b)$, so $a\sim b$ and hence $[a]=[b]$
• .
• SY: if $q$ is any element of $T$ by surjectivity of $f$ we can find $\exists d\in S$ such that $f(d)=q$, so applying $\rho(d)=[d]$, we get a class $[d]\in S/{\sim}$ with $\beta([d])=f(d)=q$
• .
• $\Box$

—-

(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. $A_5$ 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 ${\mathbb{Z}}_2*{\mathbb{Z}}_3=PSL_2({\mathbb{Z}})$
7. HNN-extensiones y $A*_CB$

### 4 responses to “álgebra”

1. atención álgebra modderna uno
http://problemhere.wordpress.com/

2. k-mat

1) group actions, class equation
2) Sylow’s theorems
3) Galois theory
4) word algebra: free groups, free products, amalgamated products…

3. <3 ÁLGEBRA <3