An answer regarding automorphisms of finite abelian groups

TL;DR

This paper investigates the possible ratios of automorphism group sizes to group order in finite abelian groups, showing restrictions on the form of these ratios.

Contribution

It proves that such ratios must have squarefree denominators and cannot equal any odd prime, answering a specific question about automorphism group sizes.

Findings

Ratios with denominator not squarefree do not occur.

No odd prime can be the ratio of automorphism group size to group order.

Provides a negative answer to a previously open question.

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 $G$.