Let $(G,•)$ be a group in which $(a•b)^3 = a^3•b^3$. Prove that $H=\{x^3 : x \in G\}$ is a normal subgroup of $G$.

Question

Let $(G,•)$ be a group in which $(a•b)^3 = a^3•b^3$. Prove that $H=\{x^3 : x \in G\}$ is a normal subgroup of $G$.

I assumed a homomorphism from $G$ to $G$ by $f(x) = x^3$. Now $\operatorname{Im}f$ is $H$. This is where I got stuck. How can I show that here $\operatorname{Im}f$ is a normal subgroup of $G$? Please help.

Answer 1

Let $a,b\in H$ so that $a=x^3$ and $b=y^3$. Since $(ab^{-1})=x^{3}y^{-3}=(xy^{-1})^3\in H$, $H$ is a subgroup.

Let $g\in G$. Then $gag^{-1}=gx^3g^{-1}=(gxg^{-1})^3\in H \implies H\lhd G$.