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Homework Help: Order of an element in a group

  1. Mar 20, 2010 #1
    Hey guys!

    I'm having some trouble trying to solve this question.. Any advice/help is appreciated!

    1. The problem statement, all variables and given/known data

    Suppose that a and b belong to a group such that:
    |a| = 4, |b| = 2, and suppose a3b=ba
    Find the order of ab.

    2. Relevant equations



    3. The attempt at a solution
    So I am unsure of which theorems I should be looking at... but anyways...... my horrible attempt:
    so a,b are in G.
    |ab| = (ab)n = 1

    Let n = order(ab), anbn = 1 --> an=b-n

    I don't know if I should continue from this point because I don't know how exactly I would go about finding the order.

    I think |ab| = |ba|, yes or no?
    If it does then I'm guessing I can use that to find the order of a3b?
    Any hints or suggestions on how?

    Thank you.
     
  2. jcsd
  3. Mar 20, 2010 #2

    Dick

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    (ab)^n and a^n*b^n are not the same thing. Clearly your group isn't abelian. Hint: (ab)(ab)=a(ba)b.
     
  4. Mar 21, 2010 #3
    Hey, thanks for the reply.
    So is this a matter of manipulating a and b ?

    For your hint, I'm just wondering why is it (ab)(ab)?
    I see that a(ba)b = a^4b^2, which we have as the original orders, but why (ab)^2 and not some other number n?
     
  5. Mar 21, 2010 #4

    Dick

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    Just because I noticed (ab)^2 worked. You can probably figure the order can't be too high with a relation like ba=a^3*b. You could try it for other n. You know n=4 will also work. The given relation tell you how to commute a and b. I.e. how to turn a 'ba' type expression into an 'ab' type expression.
     
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