Prove Group Commutativity: (G,*) w/ x*x=eG

[tasha10]

(A) Let (G,*) be a group such that x*x=eG for all x in G. Prove G is commutative.
(B) Give a specific example of an infinite group (G,*) such that x*x=eG for all x in G.

I have not gotten very far, just to let two variable x,y be in G and I know that (x*y)*(x*y) = eG .. I'm not sure where to go from here..

 

[l'Hôpital]

Well, for a, you know ...
$$(xy)^{-1} = y^{-1}x^{-1}$$

Try multiplying that to both sides of the equality you presented, and see what you get.

 

[annoymage]

hmm can i ask, what's "eG" means?? identity?

 

[tasha10]

so, will i just get eG on both sides? does this prove that it is commutative? I'm confused.

 

[tasha10]

yes, it is the identity

 

[l'Hôpital]

You won't get eG on both sides. I'm saying, multiply what I showed you to both sides of

$$xyxy = e_{G}$$