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
  • We already had studied the
  • 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

category diagram of set, binary relation and maps

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. k-mat

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

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