$\displaystyle (G;.)$ is a group.
I must show that if $\displaystyle (\forall a;b \in G): (ab)^2=a^2b^2 \Longrightarrow (G;.)$ is abelian.
Can you help me please???
And thanks.
$\displaystyle ab=ababb^{-1}a^{-1}=(ab)^2b^{-1}a^{-1}=a^2b^2b^{-1}a^{-1}=a^2ba^{-1}$ thus multiplying both sides by $\displaystyle a$ on the left gives $\displaystyle aba=a^2b\implies ba=ab$