Robert L. Griess

Framed Vertex Operator Algebras, Codes and the Moonshine Module

Abstract:For a simple vertex operator algebra whose Virasoro element is a sum of commutative Virasoro elements of central charge ½, two codes are introduced and studied. It is proved that such vertex operator algebras are rational. For lattice vertex operator algebras and related ones, decompositions into direct sums of irreducible modules for the product of the Virasoro algebras of central charge ½ are explicitly described. As an application, the decomposition of the moonshine vertex operator algebra is obtained for a distinguished system of 48 Virasoro algebras.

Finite simple groups which projectively embed in an exceptional Lie group are classified

Since finite simple groups are the building blocks of finite groups, it is natural to ask about their occurrence “in nature”. In this article, we consider their occurrence in algebraic groups and moreover discuss the general theory of finite subgroups of algebraic groups.

THE STRUCTURE OF THE “MONSTER” SIMPLE GROUP

Publisher Summary This chapter presents some evidence for the existence of a simple group F, called the monster. It was discovered independently by Fischer and Thompson. Using computer techniques, existence has been proven for F 3 and F 5 by P. Smith and S. Norton, respectively. However, the existence and uniqueness questions for F 2 and F = F 1 , remain to be settled. F has exactly two classes of involutions. F has trivial multiplier and trivial outer automorphism group. Also, F 2 has multiplier of order at most 2 and trivial outer automorphism group. The chapter also presents a few results about representations of F.

Positive definite lattices of rank at most 8

Abstract We give a short uniqueness proof for the E8 root lattice, and in fact for all positive definite unimodular lattices of rank up to 8. Our proof is done with elementary arguments, mainly these: (1) invariant theory for integer matrices; (2) an upper bound for the minimum of nonzero norms (either of the elementary bounds of Hermite or Minkowski will do). We make no use of p-adic completions, mass formulas or modular forms.

Embeddings of PSL (2,41) and PSL (2,49) in E 8 (C)

We classify embeddings of the finite simple groups PSL(2, 41) andPSL (2, 49) into algebraic groups of type E8in characteristic not dividing the orders of these respective groups. Lie theory and finite group theory point to families of elements in a group of typeE8 over a finite field which might satisfy a presentation for our finite simple group. Techniques from computational representation theory identify all suitable systems of generators. We describe a number of methods of computation in Chevalley groups of type E8.

Minimal dimensions for modular representations of the monster

It is well known that the minimal dimension of a nontrivial complex representation of the Monster is realized by Griesss module of dimension 196883. We show that the corresponding p-modular reduction also realizes the minimal dimension for character¬istics p≠2,3; but in those exceptional characteristics, involves a composition factor of the minimal dimension 196882.

Generation Three of the Happy Family and the Pariahs

We now sketch the discoveries and constructions of the remaining fourteen sporadic simple groups, and make some comments on their properties.

Assumed Results about Particular Groups

We hope that the reader has some familiarity with elementary homological algebra. We begin with a survey of elementary useful results from group extension theory. Later, we focus on particular extensions which come up in our theory of sporadic groups. For general reference, we suggest [Ben], [Gru], [HiSt], [MacL], [Rot].

Background from General Group Theory

We discuss the background in group theory used in this book and present the notation and conventions in force. Unless explicitly stated otherwise, rings are assumed to be associative with unit and modules are assumed unital. There is a summery of group theoretic notation on p.???

Subgroups of the Conway Groups; the Simple Groups of Higman-Sims, McLaughlin, Hall-Janko and Suzuki; Local Subgroups; Conjugacy Classes

We describe the involvement in CO 0 of the simple groups of Hall-Janko, Suzuki, McLaughlin and Higman-Sims. In addition, we determine conjugacy classes of elements of small order and the structure of several local subgroups (recall that a p-local subgroup, for a prime p, is the normalizer of a nonidentity p-subgroup, and a local subgroup is a p-local subgroup for some prime p). Appendix 10C contains Conway’s original table of triangle and other stabilizers, along with additional material. Our treatment of the conjugacy classes is original and our treatment of the above simple groups does involve some revision.