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Showing two groups are *Not* isomorphic

  1. Oct 2, 2015 #1
    1. The problem statement, all variables and given/known data
    Good day,

    I need to show:

    [tex]\mathbb{Z}_{4}\oplus \mathbb{Z}_{4} [/tex]is not isomorphic to [tex]\mathbb{Z}_{4}\oplus \mathbb{Z}_{2}\oplus \mathbb{Z}_{2}[/tex]

    2. Relevant equations

    None

    3. The attempt at a solution

    I was given the hint that to look at the elements of order 4 in a group. I know [tex]\mathbb{Z}_{4}\oplus \mathbb{Z}_{4} [/tex] will have the elements: (0,0)(0,1)(0,2)(0,3)(1,0)(1,1).......(3,3).

    Im a little confused on how to find the order of say (1,2) in [tex]\mathbb{Z}_{4}\oplus \mathbb{Z}_{4} [/tex].
    I know how to find the order of say <3> in [tex]\mathbb{Z}_{4}[/tex] (order=4/gcd(3,4)=4) but how can I do it with the direct sum elements?

    Thanks in advance!
     
  2. jcsd
  3. Oct 2, 2015 #2
    Can anyone tell me this is correct:

    In Z4 direct sum Z4, say we look at the element (2,0)
    Since <2> has order 2 in Z4 and <0> has order one in Z4, it follows that (2,0) has order 2 because the lcm(2,1) is 2?
     
  4. Oct 2, 2015 #3

    Dick

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    Science Advisor
    Homework Helper

    Correct. The conclusion is clearly correct since (2,0)+(2,0)=(0,0). Try some others until you can see why that rule works.
     
    Last edited: Oct 2, 2015
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