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Homework Help: Abelian group

  1. Apr 28, 2008 #1
    1. The problem statement, all variables and given/known data
    Let G be an abelian group. Show that the set of all elements of G of order 2 forms a subgroup of G. Find all elements of order 2 in Z6.

    3. The attempt at a solution
    The elements of Z6 are 1,4,5. I'm not sure how to find the set of all elements of order 2. Can someone help with that? I think I can prove from there.
     
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  3. Apr 28, 2008 #2

    Tom Mattson

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    Nope. How did you come up with that? And are you talking about the additive group [itex]\mathbb{Z}_6[/itex], or the multiplicative group [itex]\mathbb{Z}_6[/itex]? It does make a difference.
     
  4. Apr 28, 2008 #3
    We are talking about the additive group of Z6. Can you nudge me in the right direction to find the correct elements?
     
  5. Apr 28, 2008 #4

    Dick

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    If it's additive then an element g has order 2 if g+g=0 and g is not zero. It should be pretty easy to find them. But why do you say the elements are 1,4,5???
     
    Last edited: Apr 28, 2008
  6. Apr 28, 2008 #5
    OK. I understand what you are saying and know how to construct the table to find the elements of order 2. However, I still don't understand what the elements of Z6 are. Is it as obvious as 0,1,2,3,4,5?
     
  7. Apr 28, 2008 #6

    Dick

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    It's as obvious as that, yes. So for which ones is g+g=0 mod 6?
     
  8. Apr 29, 2008 #7
    (0,0), (3,3)
     
    Last edited: Apr 29, 2008
  9. Apr 29, 2008 #8
    This one is solved. Thanks for nudging me in the right direction!
     
  10. Apr 29, 2008 #9

    Dick

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    I didn't ask for which pairs is (a,b) is a+b=0. Why would I?? Look up the definition of 'order 2'. Now write it on a blackboard ten times. Now step back and look at it. Now tell me why most of the pairs in your 'guess' aren't interesting.
     
  11. Apr 29, 2008 #10

    Dick

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    Better. So the subgroup is made of the elements 0 and 3.
     
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