= Abelian image of a cubing homomorphism
{title2=$x\mapsto x^3\text{ homomorphic}\ \Longrightarrow\ \{x^3:x\in G\}\text{ abelian}$}
Suppose the cubing map $T:G\to G$ is a <group homomorphism>. Expanding $(ab)^3=a^3b^3$ and cancelling gives $(ba)^2=a^2b^2$. Also
$$
gz^3g^{-1}=(gzg^{-1})^3=g^3z^3g^{-3},
$$
so $g^2$ commutes with every cube. For $x,y\in T(G)$, therefore, $x^2y=yx^2$ and $yxyx=x^2y^2$. Multiplying the latter identity on the left by $y^{-1}$ yields $xyx=x^2y$; cancelling $x$ gives $yx=xy$. Thus the <image of a group homomorphism> $T(G)$ is an <abelian group>.
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