An answer regarding automorphisms of finite abelian groups

Abstract

In this note we provide a negative answer to the question: ``Is it true that for every positive rational number $r$ there exists a finite abelian group $G$ such that $|\mathrm{Aut}(G)|/|G|=r$?". We show that if $r=a/b$ is a rational number (with $a$ and $b$ coprime integers) so that $r=|\mathrm{Aut}(G)|/|G|$ for a finite abelian group $G$, then $b$ is squarefree. We also show that no odd prime can equal $|\mathrm{Aut}(G)|/|G|$ for a finite abelian group $G$.