1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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.
     
  2. jcsd
  3. Apr 28, 2008 #2

    Tom Mattson

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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

    User Avatar
    Science Advisor
    Homework Helper

    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

    User Avatar
    Science Advisor
    Homework Helper

    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

    User Avatar
    Science Advisor
    Homework Helper

    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

    User Avatar
    Science Advisor
    Homework Helper

    Better. So the subgroup is made of the elements 0 and 3.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Abelian group
  1. Abelian group (Replies: 1)

  2. Abelian Groups (Replies: 5)

  3. Abelian Groups (Replies: 9)

  4. Abelian group (Replies: 2)

Loading...