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