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

Abstract

The normal covering number $\gamma (G)$ of a finite group $G$ is the minimum number of proper subgroups whose conjugates cover the group. We give various estimates for $\gamma (S_n)$ and $\gamma (A_n)$ depending on the arithmetic structure of $n$. In particular we determine the limsups over $\gamma (S_n)/n$ and $\gamma (A_n)/n$ over the sequences of even and odd integers, as well as the liminf of $\gamma (S_n)/n$ over even integers. In general we explain how the values of $\gamma (S_n)/n$ and $\gamma (A_n)/n$ 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.

arXiv comment20pp. To appear in Bulletin of the London Mathematical Society

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• Permutations contained in transitive subgroups
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• A note on cube-free problems