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A group G is such than a^3 = e for every a in G. Is it abelian?

  1. Jul 17, 2012 #1
    1. The problem statement, all variables and given/known data
    Let G be a group and e its identity. This group has the property that a^3 = e, for every a in G. What I need to do is verify if this condition is sufficient for G to be abelian.

    2. The attempt at a solution
    I found a non-trivial group for which this is true, namely the group {e, a, b} with ab = ba = e, a² = b and b² = a. Other than this, I'm having a really hard time with this problem. I also tried to come up with a counter-example, but I couldn't even find another group with the property that a^3 = e for every a.
     
  2. jcsd
  3. Jul 17, 2012 #2

    Dick

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    That's actually a pretty tough question. There is a counter-example. I found it by googling. It's a matrix group with matrix entries being integers mod 3. You can probably find it too.
     
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