An Elementary Approach to the Monster

Abstract

is one of the most spectacular and mysterious mathematical achievements of the past fifty years. Unfortunately the various standard approaches to the Monster are notoriously difficult to learn. This paper presents a relatively elementary approach to the Monster. We describe a construction of the smallest non-Abelian finite simple group (of order 60), and show that it very closely parallels a construction of M x M. In section 2, before we begin our main discussion, we give a quick overview of the Monster both as a finite simple group and with respect to its most startling properties. This is done to allow those readers who have only a passing acquaintance with the Monster to get a better sense of its role in mathematics. Despite its brevity, this section has a vast scope, so we refer the reader to other sources for more comprehensive treatments (see [1], [2], or [6]). In section 3 we start with the tetrahedral graph and use Coxeter groups, along with certain additional relations, to obtain a group presentation for the finite simple group As. Then in section 4 we see that this method, when applied to different initial graphs, leads not only to the square M x M of the elusive Monster group, but to other finite simple groups as well.