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First Isomorphism Theorem Question

  1. Mar 11, 2012 #1
    I am having a hard time using or applying the theorem .

    Anyways

    Prove that there is no homomorphism from Z[itex]_{8}[/itex][itex]\oplus[/itex]Z[itex]_{2}[/itex] onto Z[itex]_{4}[/itex][itex]\oplus[/itex]Z[itex]_{4}[/itex]

    Im guessing its the First Isomorphsim Theorem because its in the chapter. But Im not sure how to use it.
     
  2. jcsd
  3. Mar 11, 2012 #2

    Dick

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    Show the kernel of the homomorphism must be nontrivial. How might you do that?
     
  4. Mar 11, 2012 #3
    just go through and try all the elements until you exhaust them all. Do I want to find the kernel of the first thing or the second thing?
     
  5. Mar 11, 2012 #4

    Dick

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    How can you do that?! You aren't given the homomorphism. Look at the orders of elements in the two groups.
     
  6. Mar 11, 2012 #5
    Cant I just make one up?
    And how do you work in the factor group?
     
  7. Mar 11, 2012 #6

    Dick

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    You can make one up if you want to. But showing that isn't onto doesn't show another homomorphism might not be onto. If the kernel is nontrivial then it can't be onto. What's the reason for that?
     
  8. Mar 11, 2012 #7
    So would the kernel be (8,0) and since (8,0)=(0,0) its not a onto homomorphism?
     
  9. Mar 11, 2012 #8

    Office_Shredder

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    I don't see any particular reason for why that would be the kernel.

    Suppose that we call the kernel of the map K. Then (Z8+Z2)/K = Z4+Z4 is an isomorphism of groups by the first isomorphism theorem. What is the cardinality of Z8+Z2/K in terms of |K|? What is the cardinality of Z4+Z4?
     
  10. Mar 11, 2012 #9

    Dick

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    That's pretty poorly worded. (8,0)=(0,0) isn't a true statement. Call the homomorphism h. Then why is h((8,0))=(0,0)?
     
  11. Mar 11, 2012 #10
    (Z8+Z2)/Z4+Z4 = K

    so |K| = 16/16=1 so wouldnt the kernel be the identity?
     
  12. Mar 11, 2012 #11
    because 8 mod 4 is zero and 0 mod 4 is zero
     
  13. Mar 11, 2012 #12

    Dick

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    There's a kernel of truth in there. Now if you would just state it in terms that involve group theory that might be helpful. Why are you talking about mod 4?
     
  14. Mar 11, 2012 #13
    we are mapping over to Z4
     
  15. Mar 11, 2012 #14

    Dick

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    No you aren't! You are mapping to Z4+Z4.
     
  16. Mar 11, 2012 #15
    yes but component wise we are mapping each thing to Z4, is that the wrong way to look at it ?
     
    Last edited: Mar 11, 2012
  17. Mar 11, 2012 #16
    Hint: One of the groups has an element of order ______ but the other one ______.
     
  18. Mar 11, 2012 #17
    aren't both groups order 16 because 8*2 =16 and 4*4=16?
     
  19. Mar 11, 2012 #18
    Wait Z8 has an element of order 8 and Z4 has an element of at most 4. Is that right.... if it is, i'm not sure if that is helpful
     
  20. Mar 11, 2012 #19

    Dick

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    It's the wrong way to look at it. You are throwing out numbers like you know what the homomorphism is and you are just cranking out arithmetic. You don't know what the homomorphism is. You can't do that. This is a course in group theory, not arithmetic. Group theory is an abstraction of arithmetic. Use group theory terminology to express yourself.
     
  21. Mar 11, 2012 #20

    Dick

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    That's very helpful. Now use it. It's your job to figure out how to use it.
     
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