Answer to Question #350793 in Abstract Algebra for liz

Question

2.7. If G is a group such that (ab)2 = a2b2 for all a, b ∈ G, then show that G must be abelian.

Answer 1

$abab=a^2b^2$ apply $a^{-1}$ from left and $b^{-1}$ from right. We obtain $ba=ab$ for all $a,b\in G$. Hence $G$ is abelian.