Let $G$ be a finite group whose order is not divisible by $3$ and such that $(ab)^3 = a^3 b^3$ for all $a$, $b$ in $G$. Then can we determine if $G$ is abelian or not?
Since $$ (ab)^3 = a^3 b^3 $$ or $$ ababab = aaabbb, $$ we can using the cancellation in $G$ write $$ baba = aabb $$ or $$ (ba)^2 = a^2 b^2. $$ What next?
And,
What if we require that $G$ be a finite group whose order is not divisible by a given positive integer $n$ and such that $(ab)^n = a^n b^n$ for all $a$, $b$ in $G$.
As before, we can write using the given condition that $$ (ba)^{n-1} = a^{n-1} b^{n-1}.$$ But What next?
