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