Anatomy of the Monster: I

Summary

Abstract

We summarise much of what is currently known about the subgroup structure of the Fischer–Griess Monster.

Introduction

The publication of the atlas of Finite Groups [1] (henceforth referred to as the atlas) marked the end of a chapter in the history of group theory in more ways than one. The classification of finite simple groups had been completed; and, using new computational techniques, the last of the sporadic groups had been constructed, and enough progress had been made in classifying their maximal subgroups to make the rest seem like a “tidying up” job.

Since the publication of the atlas, these computational methods have been pushed further. The classification of the maximal subgroups for Fi₂₂, Th, Fi₂₃J₄ and Fi′₂₄ was completed by Kleidman, Linton, Parker and Wilson in [7, 8, 9, 10, 12, 13, 14]. This means that Appendix 2 of the atlas of Brauer Characters [11], when collated with the atlas itself, completes the listing for all sporadic groups except the Baby Monster and Monster. For the Baby Monster, four new maximal subgroups ((S₆ × L₃(4):2):2, (S₆ × S₆):4, L₂(31) and L₂(49).2₃) were given in Appendix 2 of [11], and Wilson has since found four more (M₁₁, L₃(3), L₂(17):2 and L₂(ll):2); for references see [23, 24, 25]. With work on the 2-locals by Meierfrankenfeld and Shpektorov [15], an end to the Baby Monster problem is in sight.

With a lot of work also having been done on maximal subgroups of generic groups, this leaves only the Monster. Unfortunately, the degree of its smallest representation over any field, 196882, is too big for present day computational techniques to make any headway.