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arXiv · 2410.06999

Normal covering numbers for $S_n$ and $A_n$ and additive combinatorics

Sean Eberhard·Connor Mellon
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Abstract

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.

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Sean Eberhard, Connor Mellon. 2025-07-15. Normal covering numbers for $S_n$ and $A_n$ and additive combinatorics. https://doi.org/10.1112/blms.70154

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