Preface

Summary

The classification of the finite simple groups says that each finite simple group is isomorphic to exactly one of the following:

A group of prime order

An alternating group A_n of degree n

A group of Lie type

One of twenty-six sporadic groups

As a first step in the classification, each of the simple groups must be shown to exist and to be unique subject to suitable hypotheses, and the most basic properties of the group must be established. The existence of the alternating group A_n comes for free, while the representation of A_n on its n-set makes possible a first uniqueness proof and easy proofs of most properties of the group. The situation with the groups of Lie type is more difficult, but while groups of Lie rank 1 and 2 cause some problems, Lie theory provides proofs of the existence, uniqueness, and basic structure of the groups of Lie type in terms of their Lie algebras and buildings.

However, the situation with the sporadic groups is less satisfactory. Much of the existing treatment of the sporadic groups remains unpublished and the mathematics which does appear in print lacks uniformity, is spread over many papers, and often depends upon machine calculation.

Sporadic Groups represents the first step in a program to provide a uniform, self-contained treatment of the foundational material on the sporadic groups. More precisely our eventual aim is to provide complete proofs of the existence and uniqueness of the twenty-six sporadic groups subject to appropriate hypotheses, and to derive the most basic structure of the sporadics, such as the group order and the normalizers of subgroups of prime order.