1. ## Abelian Midterm question.

I had never encountered this type of question until a couple of minutes ago during a midterm...

It went something like this

Prove that If x^3=x for all x in G, then G is abelian.

I stated that since x^3=x, the only possibilities for x are 1,0 and -1. In this case I proved that all three combinations of these numbers are abelian.

Looking through some notes, should I have introduced a y with the same properties and then proven something like x^3 * y^3 =xy?

How is this proof done?

2. Originally Posted by elninio
I had never encountered this type of question until a couple of minutes ago during a midterm...

It went something like this

Prove that If x^3=x for all x in G, then G is abelian.

I stated that since x^3=x, the only possibilities for x are 1,0 and -1. In this case I proved that all three combinations of these numbers are abelian.

Looking through some notes, should I have introduced a y with the same properties and then proven something like x^3 * y^3 =xy?

How is this proof done?
You don't even know that x is a number -- so you can't say that x^3=x means $\displaystyle x=0,\pm 1$...

The correct way to go about it would be to note that $\displaystyle x^3=x \ \forall x \in G \Rightarrow x^2 = e \ \forall x \in G$, where e is the identity element in G. Now, I assume you proved the theorem that states that if every element of G is of order 2 (other than the identity obviously), then G is abelian... if you haven't, then prove that and you are done.