Introduction to Sporadic Groups for physicists

TL;DR

This paper introduces finite simple groups, especially sporadic groups, highlighting their structure and potential applications in physics, including the construction of the Monster group and connections to string theory.

Contribution

It provides an overview of finite simple groups, emphasizing sporadic groups and their relevance to physical theories and models.

Findings

Construction of the Monster group from physics arguments

Relation of Mathieu groups to string and M-theory compactifications

Identification of sporadic groups with physical applications

Abstract

We describe the collection of finite simple groups, with a view on physical applications. We recall first the prime cyclic groups $Z_p$, and the alternating groups $Alt_{n>4}$. After a quick revision of finite fields $\mathbb{F}_q$, $q = p^f$, with $p$ prime, we consider the 16 families of finite simple groups of Lie type. There are also 26 \emph{extra} "sporadic" groups, which gather in three interconnected "generations" (with 5+7+8 groups) plus the Pariah groups (6). We point out a couple of physical applications, including constructing the biggest sporadic group, the "Monster" group, with close to $10^{54}$ elements from arguments of physics, and also the relation of some Mathieu groups with compactification in string and M-theory.