The covering number of $M_{24}$

Michael Epstein,Spyros S Magliveras

doi:10.13069/jacodesmath.90728

Abstract

A finite cover $\mathcal{C}$ of a group $G$ is a finite collection of proper subgroups of $G$ such that $G$ is equal to the union of all of the members of $\mathcal{C}$. Such a cover is called {\em minimal} if it has the smallest cardinality among all finite covers of $G$. The covering number of $G$, denoted by $\sigma(G)$, is the number of subgroups in a minimal cover of $G$. In this paper the covering number of the Mathieu group $M_{24}$ is shown to be 3336.

Highlights

A finite cover C of a group G is a finite collection of proper subgroups of G such that G is equal to the union of all of the members of C
The covering number of such a group G is denoted by σ(G), and is defined by σ(G) = min{|C| : C is a finite cover of G}
As seen in [4], there are 9 conjugacy classes of maximal subgroups of M24, which we denote by Mi, 1 ≤ i ≤ 9 ordered such that |M1 | ≤ |M2 | ≤ ... ≤ |M9 |

Summary

Introduction

Abstract: A finite cover C of a group G is a finite collection of proper subgroups of G such that G is equal to the union of all of the members of C. In this paper the covering number of the Mathieu group M24 is shown to be 3336. A finite collection C of proper subgroups of a group G is said to be a finite cover of G if H∈C H = G.

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Conclusion