Show that a group $G$ is abelian if $(gh)^3 = g^3h^3$, $(gh)^4 = g^4h^4$, and $(gh)^5 = g^5h^5$ for all $g$ and $h$ in $G$.
If $G$ is a group whereby $(a\cdot b)^{i} =a^i\cdot b^i$ for three consecutive integers $i$ for all $a, b \in G$, show $G$ is abelian.