Difference between revisions of "Morphisms MA453S11Walther" - Rhea

Revision as of 16:30, 25 April 2011

For the following definitions, Let G and H be two groups: A morphism, rho, from G to H is a function rho: G --> H such that:

     1)(1G) = 1H
     2)Rho(g*gprime) = Rho(g)*Rho(gprime), this preserves the multiplication table
     The domain and the codomain are two operations that are defined on every morphism.
     Morphims satisfy two axioms:
     1)Associativity: h composed of (g composed of f) = (hcircleg)circlef whenever the operations are defined
     2)Identity: for every object X, the identity morphism on X exists such that for every morphism f: A --> B, 
       idB composed f = f = f circle idA 

Types of morphisms: An epimorphism is a morphism where for every h in H, there is at least one g in G with f(g) = h

     •This is the same as saying that rho is surjective or onto

A monomorphism is a morphism for which rho(g) = rho(gprime) can only happen if g = gprime

     •This is the same as saying that rho is injective

An isomorphism is a morphism that is both an epimorphism and a monomorphism (both surjective and injective). This means that rho sets up a 1-to-1 correspondence between the elements of G and the elements of H.

     •This is the same as saying that rho is bijective

An automorphism is an isomorphism from a function to itself. It is a way of mapping the object to itself while preserving all of its structure.

     •An inner automorphism Is a function ƒ: G → G such that ƒ(x) = a−1xa, for all x in G, where a is a given fixed element of  
      G.

A homomorphism is a structure-preserving map between two algebraic structures (such as groups, rings, or vector spaces).

     •Types of homomorphisms:
          o Group homomorphism- this is a homomorphism between two groups.
          o Ring homomorphism- this is a homomorphism between two rings.
          o Functor- this is a homomorphism between two categories
          o Linear map- this is a homomorphism between two vector spaces
          o Algebra homomorphism- this is a homomorphism between two algebras
     •Properties of elements under homomorphisms:
      Let phi be a homomorphism from a group G to a grou H and let g be and element of G. Then:
          1) Phi carries the identity of G to the identity of H
          2)Phi(g^n) = (phi(g))^n for all n in Z
          3)If |g| is finite, then |phi(g)| divides |g|
          4)Ker(phi) is a subgroup of G
          5)aKer(phi) = bKern(phi) if and only if phi(a) = phi(b)
          6)If phi(g) = gprime then phi^-1(gprime) = {x in G \ phi(x) = gprime} = gKerphi
     •Properties of Subgroups Under Homomorphisms
      Let phi be a homomorphism from a group G to a group H and let I be a subgroup of G. Then:
          1)Phi(I) = [phi(i) | i in I} is a subgroup of H
          2)If I is cyclic, then phi(I) is cyclic
          3)If I is Abelian, then phi(I) is Abelian
          4)If I is normal in G, then phi(I) is normal in phi(G)
          5)If \Kerphi\ = n, then phis is an n-to-1 mapping from G onto phi(G)
          6)If |I| = n, then |phi(I)| divides n
          7)If I bar is a subgroup of G bar, then phi^-1(I bar) = {i in G | phi(i) in Ibar} is a subgroup of G.
          8)If I bar is a normal subgroup of G bar, then phi^-1(Ibar) = {i in G\ phi(i) in Ibar} is a normal subgroup of G
          9)If phi is onto and Kerphi = {e}, then phi is an isomorphism from G to G bar.

Examples • Any isomorphism is a homomorphism that is also onto and 1-to-1 • The mapping phi from Z to Zn, definded by phi(m) = m mod n is a homomorphism • The mapping phi(x) = x^2 from R*, the nonzero real numbers under multiplication, to itself is a homomorphism. This is because phi(ab) =(ab)^2 = a^2b^2 = phi(a)phi(b) for all a and b in R* • The exponential function rho : x  e^x is an isomorphism. It is injective (monomorphism) and surjective (epimorphism) because one can take logs. • Square root: (R_t_, *)  (R_t_, *) is an isomorphism • ( *2) : Z/3Z --> Z/3Z is a monomorphism,epimorphism and isomorphism