Let $G$ be a finite group and denote by $a(n,G)$ the number of elements $g\in G$ such that $g^n=1$. Prove that if $a(n,G)\le n$ for each divisor of $|G|$, then $G$ is cyclic.
I don't know how to prove this statement. I try to use the fact that $|G|=\sum_{n |\ |G|}\varphi(n)$ (where $n$ divide $|G|$) but I do not know how to proceed with the demonstration. Any ideas or hints?
