Finite Simple Groups

Abstract

In a statistical sense, the simple groups (by which we mean, in this window, non-abelian finite simple groups) are quite rare: the Godfather of the subject has likened them to fossils, occasionally found buried among the composition factors of a general finite group [Thompson 1984]. The analogy includes the suggestion that they are hard: excavating with primitive tools may uncover some general features of a group, but will usually not reveal the deeper secrets of structure hidden within the simple factors. Prosaically, what this means is that when simple groups are present, one can only get so far with elementary group-theoretic arguments and then one gets stuck.