(New page: 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)<math>Rho</math>(1G) = 1H 2)Rho(g*gprime) = Rho(g...)
 
 
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=Morphisms=
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A student project for [[MA453]]: "Abstract Algebra"
 +
----
 
For the following definitions, Let G and H be two groups:
 
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)<math>Rho</math>(1G) = 1H
+
A '''morphism''',<math>\rho\,\!</math>, from G to H is a function <math>\rho\,\!</math>: G --> H such that:
−
       2)Rho(g*gprime) = Rho(g)*Rho(gprime), this preserves the multiplication table
+
       1)<math>I_G</math> = <math>I_H</math>
−
The domain and the codomain are two operations that are defined on every morphism.
+
       2)<math>\rho\,\!</math>(g*g') = <math>\rho\,\!</math>(g)*<math>\rho\,\!</math>(g'), this preserves the multiplication table
−
Morphims satisfy two axioms:
+
 
−
       1)Associativity: h composed of (g composed of f) = (hcircleg)circlef whenever the operations are defined
+
      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,  
 
       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
+
         <math>id_B</math> o f = f = f o <math>id_A</math>.
 +
 
 
Types of morphisms:
 
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
+
An '''epimorphism''' is a morphism where for every h in H, there is at least one g in G with f(g) =  h
−
Let phi be a homomorphism from a group G to a group H and let I be a subgroup of G. Then:
+
      •This is the same as saying that <math>\rho\,\!</math> is surjective or onto
−
1) Phi(I) = [phi(i) | i in I} is a subgroup of H
+
A '''monomorphism''' is a morphism for which <math>\rho\,\!</math>(g) = <math>\rho\,\!</math>(g') can only happen if g = g'
−
2) If I is cyclic, then phi(I) is cyclic
+
      •This is the same as saying that <math>\rho\,\!</math> is injective
−
3) If I is Abelian, then phi(I) is Abelian
+
An '''isomorphism''' is a morphism that is both an epimorphism and a monomorphism (both surjective and injective).  This means that <math>\rho\,\!</math> sets up a 1-to-1 correspondence between the elements of G and the elements of H.
−
4) If I is normal in G, then phi(I) is normal in phi(G)
+
      •This is the same as saying that <math>\rho\,\!</math> is bijective
−
5) If \Kerphi\ = n, then phis is an n-to-1 mapping from G onto phi(G)
+
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.
−
6) If |I| = n, then |phi(I)| divides n
+
      •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 
−
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.
+
      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
+
A '''homomorphism''' is a structure-preserving map between two algebraic structures (such as groups, rings, or vector spaces).
−
9) If phi is onto and Kerphi = {e}, then phi is an isomorphism from G to G bar.
+
      •Types of homomorphisms:
−
Examples
+
          o Group homomorphism- this is a homomorphism between two groups.
−
• Any isomorphism is a homomorphism that is also onto and 1-to-1
+
          o Ring homomorphism- this is a homomorphism between two rings.
−
• The mapping phi from Z to Zn, definded by phi(m) = m mod n is a homomorphism
+
          o Functor- this is a homomorphism between two categories
−
• 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*
+
          o Linear map- this is a homomorphism between two vector spaces
−
• The exponential function rho : x e^x is an isomorphism.  It is injective (monomorphism) and surjective (epimorphism) because one can take logs.  
+
          o Algebra homomorphism- this is a homomorphism between two algebras
−
• Square root: (R_t_, *) (R_t_, *) is an isomorphism
+
 
−
• ( *2) : z/2z __> Z/3Z is a monomorphism,epimorphism and isomorphism
+
      •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:
 +
          1) <math>\Phi\,\!</math> carries the identity of G to the identity of H
 +
          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|
 +
          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)
 +
          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
 +
      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 <math>\bar{G}</math>
 +
          2)If I is cyclic, then <math>\Phi\,\!</math>(I) is cyclic
 +
          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)
 +
          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
 +
          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 <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 Ker<math>\Phi\,\!</math> = {e}, then <math>\Phi\,\!</math> is an isomorphism from G to <math>\bar{G}</math>.
 +
 
 +
'''Examples'''
 +
 
 +
• Any isomorphism is a homomorphism that is also onto and 1-to-1
 +
 
 +
• 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 <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 <math>\rho\,\!</math> : x --> <math>e^x</math> is an isomorphism.  It is injective (monomorphism) and surjective (epimorphism) because one can take logs.  
 +
 
 +
 
 +
• : (<math>R_t</math> , *) --> (<math>R_t</math> , *) 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

Latest revision as of 09:16, 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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Recent Math PhD now doing a post-doctorate at UC Riverside.

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