A remark on the number of automorphisms of finite abelian groups

Abstract

Let ${\mathrm{Ab}}_0$ be the class of finite abelian groups and consider the function $f:{\mathrm{Ab}}_0\to (0,\infty )$ given by $f(G)=|\mathrm{Aut}(G)|/|(G)|$, where $\mathrm{Aut}(G)$ is the automorphism group of a finite abelian group $G$. In this short note, we prove that the image of $f$ is a dense set in $[0,\infty )$.