(10 intermediate revisions by 2 users not shown)
Line 1: Line 1:
 +
[[Category:bonus point project]]
 +
[[Category:math]]
 +
[[Category:MA453]]
 +
[[Category:algebra]]
 +
[[Category:tutorial]]
 +
 +
=Morphisms=
 +
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:
+
A '''morphism''',<math>\rho\,\!</math>, from G to H is a function <math>\rho\,\!</math>: G --> H such that:
−
       1)(1G) = 1H
+
       1)<math>I_G</math> = <math>I_H</math>
−
       2)Rho(g*gprime) = Rho(g)*Rho(gprime), this preserves the multiplication table
+
       2)<math>\rho\,\!</math>(g*g') = <math>\rho\,\!</math>(g)*<math>\rho\,\!</math>(g'), this preserves the multiplication table
  
 
       The domain and the codomain are two operations that are defined on every morphism.
 
       The domain and the codomain are two operations that are defined on every morphism.
Line 10: Line 19:
 
       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.
+
         <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
 
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
+
       •This is the same as saying that <math>\rho\,\!</math> is surjective or onto
−
A '''monomorphism''' is a morphism for which rho(g) = rho(gprime) can only happen if g = gprime
+
A '''monomorphism''' is a morphism for which <math>\rho\,\!</math>(g) = <math>\rho\,\!</math>(g') can only happen if g = g'
−
       •This is the same as saying that rho is injective
+
       •This is the same as saying that <math>\rho\,\!</math> 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.
+
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.
−
       •This is the same as saying that rho is bijective
+
       •This is the same as saying that <math>\rho\,\!</math> 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 '''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   
+
       •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.
 
       G.
 
A '''homomorphism''' is a structure-preserving map between two algebraic structures (such as groups, rings, or vector spaces).
 
A '''homomorphism''' is a structure-preserving map between two algebraic structures (such as groups, rings, or vector spaces).
Line 30: Line 39:
 
           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
 +
 
       •Properties of elements under homomorphisms:
 
       •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:
+
       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) Phi 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)Phi(g^n) = (phi(g))^n for all n in Z
+
           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 |phi(g)| divides |g|
+
           3)If |g| is finite, then |<math>\Phi\,\!</math>(g)| divides |g|
−
           4)Ker(phi) is a subgroup of G
+
           4)Ker(<math>\Phi\,\!</math>) is a subgroup of G
−
           5)aKer(phi) = bKern(phi) if and only if phi(a) = phi(b)
+
           5)aKer(<math>\Phi\,\!</math>) = bKern(<math>\Phi\,\!</math>) if and only if <math>\Phi\,\!</math>(a) = <math>\Phi\,\!</math>(b)
−
           6)If phi(g) = gprime then phi^-1(gprime) = {x in G \ phi(x) = gprime} = gKerphi
+
           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 phi be a homomorphism from a group G to a group H and let I be a subgroup of G. Then:
+
       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)Phi(I) = [phi(i) | i in I} is a subgroup of H
+
           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 phi(I) is cyclic
+
           2)If I is cyclic, then <math>\Phi\,\!</math>(I) is cyclic
−
           3)If I is Abelian, then phi(I) is Abelian
+
           3)If I is Abelian, then <math>\Phi\,\!</math>(I) is Abelian
−
           4)If I is normal in G, then phi(I) is normal in phi(G)
+
           4)If I is normal in G, then <math>\Phi\,\!</math>(I) is normal in <math>\Phi\,\!</math>(G)
−
           5)If \Kerphi\ = n, then phis is an n-to-1 mapping from G onto phi(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 |phi(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 phi^-1(I bar) = {i in G | phi(i) in Ibar} is a subgroup of G.
+
           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 phi^-1(Ibar) = {i in G\ phi(i) in Ibar} is a normal 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 phi is onto and Kerphi = {e}, then phi is an isomorphism from G to G bar.
+
           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
+
 
 +
'''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
+
• 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*
+
• 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.  
+
• 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
  
−
• Square root: (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
 +
 +
 +
'''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

Retrieved from "https://www.projectrhea.org/rhea/index.php?title=Morphisms_MA453S11Walther&oldid=54712"
Shortcuts
Help Main Wiki Page Random Page Special Pages Log in
Search

Alumni Liaison

EISL lab graduate

Mu Qiao