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

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In summary, the conversation is about verifying if a group with the property that a^3 = e for every a is sufficient for the group to be abelian. The person has found a non-trivial group that satisfies this condition and is struggling to find a counter-example. However, they eventually find a counter-example by googling.
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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.
 

1. What does it mean for a group to be abelian?

A group is abelian if the order of the elements does not affect the result of the group operation. In other words, the group operation is commutative, meaning that a*b = b*a for all elements a and b in the group.

2. What does it mean for a group to have a^3 = e for every a in G?

This means that the cube of every element in the group results in the identity element e. In other words, raising an element to the power of 3 results in the neutral element.

3. Does the property a^3 = e for every a in G imply that the group is abelian?

Yes, if a group has the property that a^3 = e for every element a, then it is abelian. This is because if a*b = c, then (a*b)^3 = c^3 = e, and by the property, (a^3)*(b^3) = e*e = e. Therefore, (a*b)^3 = a^3 * b^3 = e, which implies that a*b = b*a.

4. Are there any other properties that can determine if a group is abelian?

Yes, a group is abelian if and only if every subgroup of the group is normal. This means that the left and right cosets of the subgroup are the same, which implies that the group operation is commutative.

5. Can a group be abelian if it does not have the property a^3 = e for every a in G?

Yes, a group can still be abelian even if it does not have the property a^3 = e for every element a. This is because there are other properties that can determine if a group is abelian, as mentioned in the previous answer. However, if a group has the property a^3 = e for every a in G, then it is guaranteed to be abelian.

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