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

TL;DR

This paper investigates the normal covering numbers of symmetric and alternating groups, providing estimates and limits based on the arithmetic properties of n, and linking these to additive combinatorics problems.

Contribution

It offers new bounds and limit results for b3(S_n) and b3(A_n), connecting group covering numbers with additive combinatorics, and addresses open questions from the Kourovka Notebook.

Findings

Determined limsups of b3(S_n)/n and b3(A_n)/n for even and odd n.

Established relationships between covering numbers and additive combinatorics.

Answered key open questions in the Kourovka Notebook.

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.