This article serves as an exposition of the main result of. In we give the first computer free proof of the uniqueness of groups of type J4. In addition we supply simplified proofs of some properties of such groups, such as the structure of certain subgroups.

A group of type J4 is a finite group G possessing an involution z such that H = CG(z) satisfies F*(H) = Q is extraspecial of order 213, H/Q is isomorphic to ℝ3 extended by Aut(M22), and zG ∩ Q ≠ {z}. We prove:

Main TheoremUp to isomorphism there exists at most one group of type J4.

Janko was the first to consider groups of type J4 in, where he established various properties of such groups. For example Janko showed that each group G of type J4 is simple, he determined the order of G, and he described the normalizers of all subgroups of G of prime order. However Janko left open the question of whether there exist groups of type J4 and whether all groups of type J4 are isomorphic. Thus Janko is said to have discovered J4 and indeed J4 was the last of the 26 sporadic simple groups to be discovered.

Around 1980, Conway, Norton, Parker, and Thackray proved the existence and uniqueness of J4 using extensive machine computation. This work is discussed briefly in. Norton et al. construct J4 as a linear group in 112 dimensions over the field of order 2. While the notion of 2-local geometry did not exist at that time, this geometry plays an implicit role in.

Introduction

The combinatorial objects called buildings were first introduced (cf.) to provide a geometric approach to complex simple Lie groups – in particular the exceptional ones – and later on, more generally, to isotropic algebraic simple groups (cf. eg.). The buildings which do arise in that way are spherical, that is, have finite Weyl groups. This assertion has a partial converse, proved in: roughly speaking, there is a one-to-one correspondence between the (isomorphism classes of) buildings of irreducible spherical type and rank r ≥ 3 and the algebraic absolutely simple groups of relative rank r, where the notion of algebraic simple groups must be suitably extended so as to include, for instance, classical groups over division rings of infinite dimension over their centre. In order to have a similar statement in the rank 2 case, one must impose an extra condition, the Moufang condition, on the buildings under consideration.

The correspondence in question is established via the classification of buildings of irreducible, spherical types and rank ≥ 3. For non-spherical buildings, a full classification is known only for the affine types, in rank ≥ 4 (cf.): buildings of such a type have a “spherical building at infinity”, which is the essential tool for classification. A construction procedure for buildings of more general types given in shows that there cannot be any hope for a complete classification of buildings of arbitrary types. On the other hand, there is another wide class of groups, the Kac-Moody groups, which give rise to buildings, usually of non-spherical and non-affine types; these are “concrete” objects, which one also wishes to characterize geometrically.

Let G be a finite simple group, p a prime, S a Sylow p-subgroup of G and let {P1,…, Pn} be a minimal parabolic system of G in the sense of. Then this parabolic system does not determine G uniquely. In fact in most cases there is even an infinite group having the same parabolic system. In this paper we will make an additional assumption which will hopefully make it possible to determine G in most cases. For this approach we will use the geometry γ associated to the parabolic system (The objects are the cosets of the maximal parabolics Gi = 〈Pj ∣ ≠ i〉, i = 1, …, n and incidence iff two different cosets have nontrivial intersection).

We will consider a geometry γ (firm, residually connected) over a finite type set I. Furthermore let G be a subgroup of Aut(γ) acting flag transitively on γ. Then we define

DefinitionA thin subgeometry Δ of Γ will be called an apartment in γ with respect to G iff

1. (1) I is also the type set of Δ.

2. (2) Δ is residually connected.

3. (3) N = GΔ (the setwise stabilizer of Δ in G) acts flag transitively on Δ.

We make the following observation: Suppose γ possesses an apartment Δ with GΔ = N. Let us fix a maximal flag F of Δ. Set B = GF Then we may choose notation such that where Fi are the comaximal flags in F. As N acts flag transitively on Δ we get that for any i there is a reflection xi on Fi Now obviously 〈x1, … xn〉 acts flag transitively on Δ. So by (2) we get that N = 〈x1, … xn〉 (B ∩ N).

Introduction. The celebrated paper of Conway-Norton on connections between the Monster and certain modular functions opened up dramatic possibilities for the study of hitherto unrelated fields, but for a long time the foundations of this new subject remained obscure. Work of Borcherds and Frenkel-Lepowsky-Meurman showed that the origins of “moonshine” involved the theory of infinite-dimensional Lie algebras and that indeed the monster-modular connection could be understood in the context of two-dimensional conformal field theory. In an appendix to, Norton extended the Conway-Norton conjectures in a quite remarkable way. He expected that his new strengthened conjectures, as compelling as they were, were peculiar to the Monstrous situation and might help lead to an explanation of the genus zero problem. But these hopes were quickly dashed, in the sense that in I showed that, roughly speaking, one can associate modular forms to the elements of any finite group in such a way that the axioms which Norton introduced were satisfied. Shortly thereafter I learned that the ideas of Norton were just those which axiomatize certain algebraic aspects of “conformal field theory on an orbifold.”

Thus indeed Norton's ideas are susceptible to application to any finite group and it is my feeling that this approach will provide a suitable foundation to the subject of moonshine. In this note I want to discuss the notion of what I call an “elliptic system,” which is just a mild extension of Norton's ideas in, and show how it can be thought of as a simultaneous generalization of a modular form and of a “moonshine module.”

We present a characterization of the Monster sporadic simple group in terms of its 2-local parabolic geometry.

Introduction

We consider the largest sporadic simple group which is called the Fischer-Griess Monster or the Friendly Giant and is denoted by F1, M or FG. Here we follow the monster terminology and the notation F1.

Let G(F1) be the minimal 2-local parabolic geometry of F1 constructed in. Then G(F1) has rank 5 and belongs to a string diagram all whose nonempty edges except one are projective planes of order 2 and one terminal edge is the triple cover of the generalized quadrangle of order (2,2), related to the nonsplit extension 3·S6. The geometries having diagrams of this shape are called T-geometries.

The geometry G(F1) can be described as follows. The group H ≅ F1 contains an elementary abelian subgroup E of order 25 such that NH(E)/CH(E) ≅ L5(2). Let E1 < E2 < … < E5 = E be a chain of subgroups of E, where ∣Ei∣ = 2i, 1 ≤ i ≤ 5. Then the elements of type i in G(F1) are the subgroups of H which are conjugate to Ei; two elements are incident if one of the subgroups contains the other.

Let {α1, α2, …, α5} be a maximal flag in G(F1) and Hi be the stabilizer of αi in H. Then Hi are called the maximal parabolic subgroups associated with the action of H on G(F1). Without loss of generality we can assume that αi = Ei (clearly Hi = NH(Ei) in this case), 1 ≤ i ≤ 5. Below we present a diagram of stabilizers where under the node of type i the structure of Hi is indicated.

Abstract

This paper describes a method, algorithmic in part, for determining thoseregular maps N which are smooth covers of a given regular map M. In the case where the base map M is reflexible, the method is able to distinguish between the chiral and the reflexible covers. The method is illustrated with an important example, M=k2(O), a 9-fold covering of which is one of the two smallest chiral maps with triangular faces.

Preliminaries

The paper contains a fuller account of the following definitions and preliminary results: a map M is an embedding of a (very general) graph into a surface. We consider the map to be barycentrically subdivided into triangular regions called flags, and choose one flag to be special; this is the root flag, I. Each flag f has three neighbors, denoted fr0, fr1, fr2 as in Figure 1:

Then r0, r1, r2 are permutations on Ω, the collection of flags. These three involutions generate a group C, the connection group. A symmetry of M is a permutation of ω which commutes with C, and G(M) is the group of all symmetries. We want to discuss two kinds of regularity: (1) M is rotary if G(M) has symmetries R and S which send the root flag I to Ir1r0 and Ir1r2, respectively. G+(M) is the subgroup of G(M) generated by R and S. This is a (2,p, q) group, where p and q are the orders of R and S respectively. (2) If M is rotary, then it is reflexible provided that G(M)also contains a symmetry X which sends I to Ir1, and otherwise it is chiral.

In the last decade, very many problems about permutation groups have been resolved using the classification of finite simple groups. In this article, I describe some problems which are still open, and some fields of research which are still relatively untilled. The problems concern transitivity, order, minimal degree and base size, fixed-point-free elements, reconstruction, etc.

INTRODUCTION

Even before the classification of finite simple groups was completed in 1980, it was clear that its impact on permutation group theory would be very dramatic. By an accident of fate, I happened to be considering these matters at the time. (My thoughts then are contained in my survey paper (1981).) The succeeding decade saw this impact working through the subject, with many expected and unexpected conclusions.

Among the consequences are (in no particular order):

1. (i) the determination of the 2-transitive groups, and of the rank 3 groups;

2. (ii) the determination of primitive groups of odd degree (except for affine groups, see later);

3. (iii) primitive groups of degree n are small (order at most nc log log n) with “known” exceptions;

4. (iv) a transitive permutation group of degree greater than 1 contains a fixed-point-free element of prime-power order.

In connection with the last fact, the existence of a fixed-point-free element is trivial, and was known to Frobenius; but that it can be taken to have primepower order requires the full force of the classification. The result has an application in number theory (Fein, Kantor and Schacher 1981). However, a closely related problem, raised much earlier by Isbell (1960) in the context of game theory, is still unsolved, and shows the limits of our knowledge.

This book contains the proceedings of the L.M.S. Durham Symposium on Groups and Combinatorics, July 5–15, 1990, supported by the Science and Engineering Research Council of Great Britain.

The classification of finite simple groups was completed in 1980, and most of the conference was concerned with trends in group theory and related areas which have come to the fore since then. We have divided the material into eight sections, which we now outline.

Sporadic groups

During the conference, a spectacular proof of the ten-year-old Conway “Y555 conjecture” concerning a Coxeter-type presentation of the Monster sporadic group, was achieved by Norton and Ivanov. In their articles, Norton and Ivanov give details of their proof; Ivanov obtains a new geometric characterization of the Monster, and this is used by Norton to prove the conjecture. Conway's short paper outlines the background. The article of Conway and Pritchard gives a proof of the Y552 presentation for the Fischer group Fi24. In a different vein, the papers of Aschbacher and Segev concern proofs of uniqueness results for sporadic groups. In their first article they survey their general framework for providing uniqueness results; this uses a graph-theoretical setting and some ideas with a topological flavour. Their second paper discusses in detail the uniqueness proof for the group J4.

Moonshine

This section concerns the remarkable relationships between the Monster group and certain modular functions. Much insight into moonshine is gained from studying the Monster Lie algebra, which is described by Borcherds in his article.