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Please show that Hom(Zn,Zn)=Zn.

  1. Apr 7, 2012 #1
    Please show that Hom(Z/nZ,Z/nZ) isomorphic to Z/nZ.
    Thanks..
     
  2. jcsd
  3. Apr 7, 2012 #2

    morphism

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    Well, what are your thoughts on the problem?
     
  4. Apr 8, 2012 #3
    Build a h: Hom(Z/nZ,Z/nZ) --> Z/nZ homomorphism determined by h(f)=f(1). Is it true?
     
  5. Apr 8, 2012 #4
    Go from here and check whether it is well defined (i.e. actually a homomorphism), whether it is in fact injective and surjective (sometimes called one-to-one and onto) then determine whether the inverse is also a homomorphism.
     
  6. Apr 8, 2012 #5
    Actually, it is not necessary to check that the homomorphism is injective or surjective: he's asking for Hom(Z/nZ,Z/nZ), not Aut(Z/nZ), so the maps need not be isomorphisms.
     
  7. Apr 8, 2012 #6

    Office_Shredder

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    He needs to check that house map from hom to Zn is injective and surjective
     
  8. Apr 8, 2012 #7
    Exactly the map h mentioned above does actually need to be an isomorphism thus injective and surjective. Also the inverse map needs to be a homomorphism.

    The elements of Hom(Z/nZ, Z/nZ) do not need to be isomorphisms.

    The trick here is of course to see that homomorphism are uniquely defined by their image of 1 To show this you need to use the fact that they are homomorphisms
     
  9. Apr 8, 2012 #8
    Oops, I didn't read carefully enough! Yes, the map h:Hom(Z/nZ,Z/nZ)--->Z/nZ must be an isomorphism; I was thinking of the elements of Hom(Z/nZ,Z/nZ), which of course don't have to be isomorphisms.
     
  10. Apr 8, 2012 #9
    Yes the elements of hom dont need to be isomorphism just being homomorphism is enough. And i think i found proof.
     
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