The Covering Numbers of the McLaughlin Group and some Primitive Groups of Low Degree

TL;DR

This paper investigates the minimal covers and covering numbers of the McLaughlin sporadic simple group and certain low degree primitive groups, contributing to the understanding of their subgroup structures.

Contribution

It provides new results on the covering numbers of the McLaughlin group and specific primitive groups of low degree, expanding knowledge of their subgroup coverings.

Findings

Determined the covering number of the McLaughlin group.

Established bounds for covering numbers of certain primitive groups.

Identified minimal covers for these groups.

Abstract

A \emph{finite cover} of a group $G$ is a finite collection $\mathcal{C}$ of proper subgroups of $G$ with the property that $\bigcup \mathcal{C} = G$. A finite group admits a finite cover if and only if it is noncyclic. More generally, it is known that a group admits a finite cover if and only if it has a finite, noncyclic homomorphic image. If $\mathcal{C}$ is a finite cover of a group $G$, and no cover of $G$ with fewer subgroups exists, then $\mathcal{C}$ is said to be a \emph{minimal cover} of $G$, and the cardinality of $\mathcal{C}$ is called the \emph{covering number} of $G$, denoted by $\sigma(G)$. Here we investigate the covering numbers of the McLaughlin sporadic simple group and some low degree primitive groups.