Introduction to Sporadic Groups

TL;DR

This paper introduces finite simple groups, especially sporadic groups, for physicists, detailing their classification, structure, and some physical applications, including recent examples involving sporadic groups.

Contribution

It provides a structured overview of sporadic groups, their classification, and discusses their potential applications in physics, making the topic accessible to physicists.

Findings

Classification of all sporadic groups and their levels

Description of physical applications of sporadic groups

Inclusion of recent examples using sporadic groups

Abstract

This is an introduction to finite simple groups, in particular sporadic groups, intended for physicists. After a short review of group theory, we enumerate the $1+1+16=18$ families of finite simple groups, as an introduction to the sporadic groups. These are described next, in three levels of increasing complexity, plus the six isolated "pariah" groups. The (old) five Mathieu groups make up the first, smallest order level. The seven groups related to the Leech lattice, including the three Conway groups, constitute the second level. The third and highest level contains the Monster group $\mathbb M$, plus seven other related groups. Next a brief mention is made of the remaining six pariah groups, thus completing the $5+7+8+6=26$ sporadic groups. The review ends up with a brief discussion of a few of physical applications of finite groups in physics, including a couple of recent examples which use sporadic groups.