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

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SUMMARY

The discussion centers on the group property where \( a^3 = e \) for every element \( a \) in group \( G \). A non-trivial example provided is the group \( \{e, a, b\} \) where \( ab = ba = e \), \( a^2 = b \), and \( b^2 = a \). The participant struggled to find additional examples or counter-examples but identified a matrix group with integer entries mod 3 as a counter-example. This indicates that the property \( a^3 = e \) does not guarantee that \( G \) is abelian.

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Homework Statement


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.
 
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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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