A remark on the number of automorphisms of finite abelian groups

TL;DR

This paper studies the automorphism groups of finite abelian groups and proves that the ratio of automorphisms to group order is dense in the positive real numbers.

Contribution

It establishes that the set of ratios of automorphism group sizes to group orders for finite abelian groups is dense in the positive reals.

Findings

The ratio $f(G)$ is dense in $[0, \infty)$ for finite abelian groups.

Automorphism group sizes vary widely relative to group order.

The result provides insight into the diversity of automorphism structures in finite abelian groups.

Abstract

Let $Ab_0$ be the class of finite abelian groups and consider the function $f:Ab_0\longrightarrow(0,\infty)$ given by $f(G)=\frac{|{\rm Aut}(G)|}{|G|}$\,, where ${\rm Aut}(G)$ 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 $[0,\infty)$.