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Determine is endomorphism, kernel, epimorphism, monomorphism

  1. Jul 29, 2011 #1
    I'm reasonably certain I did 1b correctly. I'm not sure about 1h. In both cases, since phi is the transformation from the multiplicative group G to the multiplicative group G, the identity is 1.

    http://i111.photobucket.com/albums/n149/camarolt4z28/IMG_20110729_200311.jpg?t=1311988303 [Broken]

    http://i111.photobucket.com/albums/n149/camarolt4z28/IMG_20110729_200408.jpg?t=1311988318 [Broken]

    http://i111.photobucket.com/albums/n149/camarolt4z28/IMG_20110729_200704.jpg?t=1311988336 [Broken]
     
    Last edited by a moderator: May 5, 2017
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  3. Jul 29, 2011 #2

    micromass

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    Seems to be entirely correct! :smile:

    One small detail, though. You checked that 1b was a monomorphism directly with the definition. This is unnecessary work as you also calculated the kernel. Knowing that the kernel is {1} is good enough for showing it's a mono.
     
  4. Jul 29, 2011 #3

    SammyS

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    Yup, looks good.
     
  5. Jul 29, 2011 #4
    Thanks for checking my work.

    How does the kernel = {1} show one-to-one?
     
  6. Jul 29, 2011 #5
    Thanks.

    According to the definition, a homomorphism is an endomorphism if the mapping is to itself, i.e. G = G'. So, in this problem, since G = G (R - {0}), you just have to show it's a homomorphism or not.
     
  7. Jul 29, 2011 #6
    You could try showing this! It only requires the definition of a group homomorphism. And it's actually both a necessary and sufficient condition (it'll be an "if and only if" proof).
     
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