Prove that a group is abelian. [duplicate]
HINT: You need to prove $ab = ba$, $\forall a,b \in G$. Note that since $ab \in G$ we also have that $(ab)^2 = e$ i.e. $(ab)(ab) = e$. Further, $a^{-1} = a$ and $b^{-1} = b$. Can you now finish this off?
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 $abab=e$.
If $a*a=e$ then $a=a^{-1}$. It follows that $(a*b)^{-1}=a*b$ but $a*b= b^{-1}*a^{-1}$ you get the desired equality for comparing both expressions.