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 $G$.

Theorems & Definitions (10)

• Theorem 1
• Theorem 2
• Theorem 3
• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• Proposition 2.4
• proof
• Proposition 2.5
• proof