Normal covering numbers for $S_n$ and $A_n$ and additive combinatorics
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913968804593664 |
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| author | Eberhard, Sean Mellon, Connor |
| author_facet | Eberhard, Sean Mellon, Connor |
| contents | The normal covering number $γ(G)$ of a finite group $G$ is the minimum number of proper subgroups whose conjugates cover the group. We give various estimates for $γ(S_n)$ and $γ(A_n)$ depending on the arithmetic structure of $n$. In particular we determine the limsups over $γ(S_n) / n$ and $γ(A_n) / n$ over the sequences of even and odd integers, as well as the liminf of $γ(S_n) / n$ over even integers. In general we explain how the values of $γ(S_n) / n$ and $γ(A_n) / n$ are related to problems in additive combinatorics. These results answer most of the questions posed by Bubboloni, Praeger, and Spiga as Problem 20.17 of the Kourovka Notebook. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_06999 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Normal covering numbers for $S_n$ and $A_n$ and additive combinatorics Eberhard, Sean Mellon, Connor Group Theory Combinatorics 20B35, 20B30, 11P70 The normal covering number $γ(G)$ of a finite group $G$ is the minimum number of proper subgroups whose conjugates cover the group. We give various estimates for $γ(S_n)$ and $γ(A_n)$ depending on the arithmetic structure of $n$. In particular we determine the limsups over $γ(S_n) / n$ and $γ(A_n) / n$ over the sequences of even and odd integers, as well as the liminf of $γ(S_n) / n$ over even integers. In general we explain how the values of $γ(S_n) / n$ and $γ(A_n) / n$ are related to problems in additive combinatorics. These results answer most of the questions posed by Bubboloni, Praeger, and Spiga as Problem 20.17 of the Kourovka Notebook. |
| title | Normal covering numbers for $S_n$ and $A_n$ and additive combinatorics |
| topic | Group Theory Combinatorics 20B35, 20B30, 11P70 |
| url | https://arxiv.org/abs/2410.06999 |