Alg7.16

From Exampleproblems

Let $G$ be a group such that ( $a b) 2 = a 2 b 2$ for all $a,b\in G$ . Prove $G$ is abelian.

$abab=(ab)^2=a^2b^2 \Rightarrow a^{-1}(abab)b^{-1}=a^{-1}(a^2b^2)b^{-1} \Rightarrow ba=ab\,$

Therefore, every $a\in G$ commutes with every other element of the group.