Abelian image of a cubing homomorphism ($x\mapsto x^3\text{ homomorphic}⟹\{x^3:x\in G\}\text{ abelian}$)

Suppose the cubing map $T:G\to G$ is a group homomorphism. Expanding $(ab{)}^3=a^3{b}^3$ and cancelling gives $(ba{)}^2=a^2{b}^2$. Alsoso $g^2$ commutes with every cube. For $x,y\in T(G)$, therefore, $x^{2}y=y{x}^2$ and $yxyx=x^2{y}^2$. Multiplying the latter identity on the left by $y^{-1}$ yields $xyx=x^{2}y$; cancelling $x$ gives $yx=xy$. Thus the image of a group homomorphism $T(G)$ is an abelian group.