On the Covering Number of Small Symmetric Groups and Some Sporadic Simple Groups

TL;DR

This paper determines the covering numbers of small symmetric groups and some sporadic simple groups, advancing understanding of subgroup coverings in finite group theory.

Contribution

It explicitly computes the covering numbers for certain small symmetric groups and sporadic simple groups, filling gaps in known data.

Findings

Calculated (S_8), (S_9), (S_{10}), (S_{12})

Established (M_{12})

Improved bounds for Janko group J_1

Abstract

A set of proper subgroups is a covering for a group if its union is the whole group. The minimal number of subgroups needed to cover $G$ is called the covering number of $G$, denoted by $\sigma(G)$. Determining $\sigma(G)$ is an open problem for many non-solvable groups. For symmetric groups $S_n$, Mar\'oti determined $\sigma(S_n)$ for odd $n$ with the exception of $n=9$ and gave estimates for $n$ even. In this paper we determine $\sigma(S_n)$ for $n = 8$, $9$, $10$ and $12$. In addition we find the covering number for the Mathieu group $M_{12}$ and improve an estimate given by Holmes for the Janko group $J_1$.