THE 68th ROMANIAN MATHEMATICAL OLYMPIAD

Since $f$ is a morphism, $(xy{)}^{m+1}=x^{m+1}{y}^{m+1}$, so $(yx{)}^m=x^m{y}^m$ for all $x$ and all $y$ in $G$. Further, $y^{m+1}{x}^{m+1}=(yx{)}^{m+1}=(yx{)}^m(yx)=(x^m{y}^m)(yx)=x^m{y}^{m+1}x$, so $y^{m+1}{x}^m=x^m{y}^{m+1}$. Since $f$ is surjective, the latter shows that $x^m$ is in the center of $G$. Similarly, $x^n$ is in the center of $G$. Finally, since $m$ is coprime to $n$, it follows that every element of $G$ is in the center, so $G$ is commutative.