An answer regarding automorphisms of finite abelian groups
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912994535931904 |
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| author | McCulloch, Ryan |
| author_facet | McCulloch, Ryan |
| contents | 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 $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_29299 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An answer regarding automorphisms of finite abelian groups McCulloch, Ryan Group Theory Primary 20D15, Secondary 20D45 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 $G$. |
| title | An answer regarding automorphisms of finite abelian groups |
| topic | Group Theory Primary 20D15, Secondary 20D45 |
| url | https://arxiv.org/abs/2603.29299 |