Multiplicative groups avoiding a fixed group
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866910598924599296 |
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| author | Hannesson, Matthias Martin, Greg |
| author_facet | Hannesson, Matthias Martin, Greg |
| contents | We know that any finite abelian group $G$ appears as a subgroup of infinitely many multiplicative groups $\mathbb{Z}_n^\times$ (the abelian groups of size $ϕ(n)$ that are the multiplicative groups of units in the rings $\mathbb{Z}/n\mathbb{Z}$). It seems to be less well appeciated that $G$ appears as a subgroup of almost all multiplicative groups $\mathbb{Z}_n^\times$. We exhibit an asymptotic formula for the counting function of those integers whose multiplicative group fails to contain a copy of $G$, for all finite abelian groups $G$ (other than the trivial one-element group). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_06869 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Multiplicative groups avoiding a fixed group Hannesson, Matthias Martin, Greg Number Theory 11N25, 11N37, 11N45, 11N64, 20K01 We know that any finite abelian group $G$ appears as a subgroup of infinitely many multiplicative groups $\mathbb{Z}_n^\times$ (the abelian groups of size $ϕ(n)$ that are the multiplicative groups of units in the rings $\mathbb{Z}/n\mathbb{Z}$). It seems to be less well appeciated that $G$ appears as a subgroup of almost all multiplicative groups $\mathbb{Z}_n^\times$. We exhibit an asymptotic formula for the counting function of those integers whose multiplicative group fails to contain a copy of $G$, for all finite abelian groups $G$ (other than the trivial one-element group). |
| title | Multiplicative groups avoiding a fixed group |
| topic | Number Theory 11N25, 11N37, 11N45, 11N64, 20K01 |
| url | https://arxiv.org/abs/2409.06869 |