# Abstract Algebra Group Abelian Proof

If $(ab)^{2}=a^{2}b^{2}$ in a group, then $ab=ba$.

Proof:
Since $(ab)^{2}=a^{2}b^{2}$ then $(ab)(ab)=aabb$. Then $ababb^{-1}=aabbb^{-1}$ and so $aba=aab$. Then $a^{-1}aba=a^{-1}aab$. So $ba=ab$.
QED