Prove that a group is abelian. 
Possible Duplicate:
Group where every element is order 2 


Let $(G,\star)$ be a group with identity element $e$ such that $a \star a = e$ for all $a \in G$. Prove that $G$ is abelian.

Ok, what i got is this: we want to prove that ab=ba, i.e. if aa=e , a=a' where a' is the inverse and bb=e, b=b' where b' is the inverse so ab=(ab)'=b'a'=ba.... 

Given $a,b\in G$ we want to know $ab=ba$, i.e. $aba^{-1}b^{-1}=e$.
What we know is that $a*a=e$, i.e. $a=a^{-1}$, similarly $b=b^{-1}$, so what we need to verify is that $abab=e$.  
