Problem

Source: Romanian National Olympiad, Grade XII

Tags: abstract algebra, superior algebra, group theory, Sylow Subgroup, college contests



Let $G$ be a finite group with the following property: If $f$ is an automorphism of $G$, then there exists $m\in\mathbb{N^\star}$, so that $f(x)=x^{m} $ for all $x\in G$. Prove that G is commutative. Marian Andronache