On the covering number of symmetric groups of even degree

TL;DR

This paper determines the exact covering number of symmetric groups of even degree divisible by 6, building on prior asymptotic bounds and extending the understanding of group covers.

Contribution

It provides the precise value of the covering number for symmetric groups when the degree is divisible by 6, filling a gap in the existing literature.

Findings

Exact covering number for $S_n$ when $n$ is divisible by 6

Extension of previous asymptotic bounds to exact values

Improved understanding of subgroup covers in symmetric groups

Abstract

If a group $G$ is the union of proper subgroups $H_1, \dots, H_k$, we say that the collection $\{H_1, \dots H_k \}$ is a cover of $G$, and the size of a minimal cover (supposing one exists) is the covering number of $G$, denoted $\sigma(G)$. Mar\'oti showed that $\sigma(S_n) = 2^{n-1}$ for $n$ odd and sufficiently large, and he also gave asymptotic bounds for $n$ even. In this paper, we determine the exact value of $\sigma(S_n)$ when $n$ is divisible by $6$.