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=Morphisms=
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A student project for [[MA453]]: "Abstract Algebra"
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----
 
For the following definitions, Let G and H be two groups:
 
For the following definitions, Let G and H be two groups:
  
 
A '''morphism''',<math>\rho\,\!</math>, from G to H is a function <math>\rho\,\!</math>: G --> H such that:
 
A '''morphism''',<math>\rho\,\!</math>, from G to H is a function <math>\rho\,\!</math>: G --> H such that:
       1)(1G) = 1H
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       1)<math>I_G</math> = <math>I_H</math>
 
       2)<math>\rho\,\!</math>(g*g') = <math>\rho\,\!</math>(g)*<math>\rho\,\!</math>(g'), this preserves the multiplication table
 
       2)<math>\rho\,\!</math>(g*g') = <math>\rho\,\!</math>(g)*<math>\rho\,\!</math>(g'), this preserves the multiplication table
  
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       1)Associativity: h o (g o f) = (h o g)o f whenever the operations are defined
 
       1)Associativity: h o (g o f) = (h o g)o f 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,  
 
       2)Identity: for every object X, the identity morphism on X exists such that for every morphism f: A --> B,  
         id_B_ o f = f = f o idA.
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         <math>id_B</math> o f = f = f o <math>id_A</math>.
  
 
Types of morphisms:
 
Types of morphisms:
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           o Linear map- this is a homomorphism between two vector spaces
 
           o Linear map- this is a homomorphism between two vector spaces
 
           o Algebra homomorphism- this is a homomorphism between two algebras
 
           o Algebra homomorphism- this is a homomorphism between two algebras
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       •Properties of elements under homomorphisms:
 
       •Properties of elements under homomorphisms:
 
       Let <math>\Phi\,\!</math> be a homomorphism from a group G to a grou H and let g be and element of G. Then:
 
       Let <math>\Phi\,\!</math> be a homomorphism from a group G to a grou H and let g be and element of G. Then:
 
           1) <math>\Phi\,\!</math> carries the identity of G to the identity of H
 
           1) <math>\Phi\,\!</math> carries the identity of G to the identity of H
           2)<math>\Phi\,\!</math>(g^n) = (<math>\Phi\,\!</math>(g))^n for all n in Z
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           2)<math>\Phi\,\!</math>(<math>g^n</math>) = (<math>\Phi\,\!</math><math>(g))^n</math> for all n in Z
 
           3)If |g| is finite, then |<math>\Phi\,\!</math>(g)| divides |g|
 
           3)If |g| is finite, then |<math>\Phi\,\!</math>(g)| divides |g|
 
           4)Ker(<math>\Phi\,\!</math>) is a subgroup of G
 
           4)Ker(<math>\Phi\,\!</math>) is a subgroup of G
 
           5)aKer(<math>\Phi\,\!</math>) = bKern(<math>\Phi\,\!</math>) if and only if <math>\Phi\,\!</math>(a) = <math>\Phi\,\!</math>(b)
 
           5)aKer(<math>\Phi\,\!</math>) = bKern(<math>\Phi\,\!</math>) if and only if <math>\Phi\,\!</math>(a) = <math>\Phi\,\!</math>(b)
           6)If <math>\Phi\,\!</math>(g) = g' then <math>\Phi\,\!</math>^-1(gprime) = {x in G | <math>\Phi\,\!</math>(x) = g'} = gKer<math>\Phi\,\!</math>
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           6)If <math>\Phi\,\!</math>(g) = g' then <math>\Phi\,\!</math>(g') = {x in G | <math>\Phi\,\!</math>(x) = g'} = gKer<math>\Phi\,\!</math>
 
       •Properties of Subgroups Under Homomorphisms
 
       •Properties of Subgroups Under Homomorphisms
       Let <math>\Phi\,\!</math> be a homomorphism from a group G to a group H and let I be a subgroup of G. Then:
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       Let <math>\Phi\,\!</math> be a homomorphism from a group G to a group <math>\bar{G}</math> and let I be a subgroup of G. Then:
           1)<math>\Phi\,\!</math>(I) = [<math>\Phi\,\!</math>(i) | i in I} is a subgroup of H
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           1)<math>\Phi\,\!</math>(I) = [<math>\Phi\,\!</math>(i) | i in I} is a subgroup of <math>\bar{G}</math>
 
           2)If I is cyclic, then <math>\Phi\,\!</math>(I) is cyclic
 
           2)If I is cyclic, then <math>\Phi\,\!</math>(I) is cyclic
 
           3)If I is Abelian, then <math>\Phi\,\!</math>(I) is Abelian
 
           3)If I is Abelian, then <math>\Phi\,\!</math>(I) is Abelian
 
           4)If I is normal in G, then <math>\Phi\,\!</math>(I) is normal in <math>\Phi\,\!</math>(G)
 
           4)If I is normal in G, then <math>\Phi\,\!</math>(I) is normal in <math>\Phi\,\!</math>(G)
           5)If |Ker<math>\Phi\,\!</math>| = n, then phis is an n-to-1 mapping from G onto <math>\Phi\,\!</math>(G)
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           5)If |Ker<math>\Phi\,\!</math>| = n, then <math>\Phi\,\!</math> is an n-to-1 mapping from G onto <math>\Phi\,\!</math>(G)
 
           6)If |I| = n, then |<math>\Phi\,\!</math>(I)| divides n
 
           6)If |I| = n, then |<math>\Phi\,\!</math>(I)| divides n
           7)If I bar is a subgroup of G bar, then <math>\Phi\,\!</math>^-1(I bar) = {i in G | <math>\Phi\,\!</math>(i) in Ibar} is a subgroup of G.
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           7)If <math>\bar{I}</math> is a subgroup of <math>\bar{G}</math>, then <math>\Phi\,\!</math>^-1(<math>\bar{I}</math>) = {i in G | <math>\Phi\,\!</math>(i) in <math>\bar{I}</math>} is a subgroup of G.
           8)If I bar is a normal subgroup of G bar, then <math>\Phi\,\!</math>^-1(Ibar) = {i in G| <math>\Phi\,\!</math>(i) in Ibar} is a normal subgroup of G
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           8)If <math>\bar{I}</math> is a normal subgroup of <math>\bar{G}</math>, then <math>\Phi\,\!</math>^-1(<math>\bar{I}</math>) = {i in G| <math>\Phi\,\!</math>(i) in <math>\bar{I}</math>} is a normal subgroup of G
           9)If <math>\Phi\,\!</math> is onto and Kerphi = {e}, then <math>\Phi\,\!</math> is an isomorphism from G to G bar.
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           9)If <math>\Phi\,\!</math> is onto and Ker<math>\Phi\,\!</math> = {e}, then <math>\Phi\,\!</math> is an isomorphism from G to <math>\bar{G}</math>.
Examples
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'''Examples'''
  
 
• Any isomorphism is a homomorphism that is also onto and 1-to-1
 
• 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
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• The mapping <math>\Phi\,\!</math> from Z to <math>Z_n</math>, definded by <math>\Phi\,\!</math>(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*
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• The mapping <math>\Phi\,\!</math>(x) = <math>x^2</math> from R*, the nonzero real numbers under multiplication, to itself is a homomorphism.  This is because <math>\Phi\,\!</math>(ab) =<math>(ab)^2</math> = <math>a^2b^2</math> = <math>\Phi\,\!</math>(a)<math>\Phi\,\!</math>(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.  
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• The exponential function <math>\rho\,\!</math> : x --> <math>e^x</math> is an isomorphism.  It is injective (monomorphism) and surjective (epimorphism) because one can take logs.  
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• : (<math>R_t</math> , *) --> (<math>R_t</math> , *) is an isomorphism
  
• : (R_t_, *) --> (R_t_, *) is an isomorphism
 
  
 
• ( *2) : Z/3Z --> Z/3Z is a monomorphism,epimorphism and isomorphism
 
• ( *2) : Z/3Z --> Z/3Z is a monomorphism,epimorphism and isomorphism
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'''Links to University Pages'''
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• http://astarmathsandphysics.com/university_maths_notes/abstract_algebra_and_group%20theory/university_maths_notes_abstract_algebra_group_theory_morphisms.html
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• http://www.math.purdue.edu/~mdd/Publications/Qd-morphisms-JFA.pdf
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• http://www.math.purdue.edu/~lipshitz/cexprintss.pdf
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• http://www.math.purdue.edu/~mdd/Publications/shape.pdf
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• http://www.math.purdue.edu/~mdd/Publications/A.pdf
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• https://www.projectrhea.org/rhea/index.php/NotesWeek4Th_MA453Fall2008walther
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'''Other interesting pages on Morphisms'''
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• http://www.jstor.org/stable/3481861?seq=2
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• http://www.math.columbia.edu/~scautis/papers/pmm.pdf
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'''References'''
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• http://en.wikipedia.org/wiki/Morphism
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• Gallian, J.A. (2010). Contemporary Abstract Algebra. United States: Brooks/Cole. 
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• MA 453 class notes, Professor Walther Lecture

Latest revision as of 10:16, 21 March 2013


Morphisms

A student project for MA453: "Abstract Algebra"


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)$ I_G $ = $ I_H $
     2)$ \rho\,\! $(g*g') = $ \rho\,\! $(g)*$ \rho\,\! $(g'), 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 o (g o f) = (h o g)o f 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, 
       $ id_B $ o f = f = f o $ id_A $.

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\,\! $(g') can only happen if g = g'

     •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) = g' then $ \Phi\,\! $(g') = {x in G | $ \Phi\,\! $(x) = g'} = gKer$ \Phi\,\! $
     •Properties of Subgroups Under Homomorphisms
      Let $ \Phi\,\! $ be a homomorphism from a group G to a group $ \bar{G} $ and let I be a subgroup of G. Then:
          1)$ \Phi\,\! $(I) = [$ \Phi\,\! $(i) | i in I} is a subgroup of $ \bar{G} $
          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\,\! $(G)
          5)If |Ker$ \Phi\,\! $| = n, then $ \Phi\,\! $ is an n-to-1 mapping from G onto $ \Phi\,\! $(G)
          6)If |I| = n, then |$ \Phi\,\! $(I)| divides n
          7)If $ \bar{I} $ is a subgroup of $ \bar{G} $, then $ \Phi\,\! $^-1($ \bar{I} $) = {i in G | $ \Phi\,\! $(i) in $ \bar{I} $} is a subgroup of G.
          8)If $ \bar{I} $ is a normal subgroup of $ \bar{G} $, then $ \Phi\,\! $^-1($ \bar{I} $) = {i in G| $ \Phi\,\! $(i) in $ \bar{I} $} is a normal subgroup of G
          9)If $ \Phi\,\! $ is onto and Ker$ \Phi\,\! $ = {e}, then $ \Phi\,\! $ is an isomorphism from G to $ \bar{G} $.

Examples

• Any isomorphism is a homomorphism that is also onto and 1-to-1

• The mapping $ \Phi\,\! $ from Z to $ Z_n $, 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.


• : ($ R_t $ , *) --> ($ R_t $ , *) is an isomorphism


• ( *2) : Z/3Z --> Z/3Z is a monomorphism,epimorphism and isomorphism


Links to University Pages

http://astarmathsandphysics.com/university_maths_notes/abstract_algebra_and_group%20theory/university_maths_notes_abstract_algebra_group_theory_morphisms.html

http://www.math.purdue.edu/~mdd/Publications/Qd-morphisms-JFA.pdf

http://www.math.purdue.edu/~lipshitz/cexprintss.pdf

http://www.math.purdue.edu/~mdd/Publications/shape.pdf

http://www.math.purdue.edu/~mdd/Publications/A.pdf

https://www.projectrhea.org/rhea/index.php/NotesWeek4Th_MA453Fall2008walther

Other interesting pages on Morphisms

http://www.jstor.org/stable/3481861?seq=2

http://www.math.columbia.edu/~scautis/papers/pmm.pdf

References

http://en.wikipedia.org/wiki/Morphism

• Gallian, J.A. (2010). Contemporary Abstract Algebra. United States: Brooks/Cole.

• MA 453 class notes, Professor Walther Lecture

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