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Need help proving a group is abelian

  1. Oct 13, 2011 #1
    I have a midterm tomorrow morning and I am completely lost on how to finish the problem, I was told a question tomorrow will mirror this one so any help is appreciated.


    Prove any group of order 9 is abelian.


    Let G be a group such that |G|=9

    One of these elements has to be the identity.

    The remaining 8 will consist of 4 elements and their respective inverses.

    Where do I go from here?
  2. jcsd
  3. Oct 13, 2011 #2
    perhaps making a cayley table will help.
  4. Oct 13, 2011 #3
    How do I go about creating a cayley table?
  5. Oct 13, 2011 #4
    You put all the elements, presumable a, b, c, d, e, f, g, h, i in a table, across and down (like a multiplication table) and then fill in a*a=? a*b=?

    But for nine elements this may not be the best way to approach this problem.
  6. Oct 13, 2011 #5
    What is another way of approaching it without constructing the tables?
  7. Oct 13, 2011 #6


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    Science Advisor
    Homework Helper

    If |G|=9, then if G has an element of order 9, then it's a cyclic group and it's abelian. Problem solved. If not then all nonidentity elements of G must have order 3, right? Start from there.
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