Finite Simple Groups: Thirty Years of the Atlas and Beyond Celebrating the Atlases and Honoring John Conway

Princeton University, November 2-5, 2015

TITLES AND ABSTRACTS

Michael Aschbacher (Caltech)

The structure of finite groups

Many problems in finite can be reduced to the case where the group G in question is nearly simple. Then to solve the problem, one needs strong information about the subgroup structure of G. I’ll talk about such reductions for permutation groups, and say some things about how one describes the subgroup structure of almost simple groups. Finally I’ll illustrate with an open question from universal algebra. The talk will be expository.

Michel Brou´e(Paris 7)

A generic Atlas for Spetses ?

In the Spetses program, the construction of an Atlas precedes the knowledge of the objects. We shall try to present part of what is computed, known, dreamt about Spetses.

Jon Carlson (U. Georgia)

Endotrivial modules

Let G be a finite group and k a field of characteristic p, dividing the of G.A kG-module is endotrivial if its endomorphism ring Homk(M,M), as a kG-module, is isomorphic to k in the stable category, stmod(kG). Tensoring with an endotrivial module is a self-equivalence of Morita type on the stable category, stmod(kG), of finitely generated kG-modules modulo projectives. The endotrivial modules have been classified in the case that G is a p-group and in many other cases. This lecture will present a survey of some recent developments involving the work of several people.

John Conway (Princeton)

(TBA)

Rob Curtis (Birmingham)

The Thompson chain of of the Co1 and complete graphs on n vertices.

The large Conway Co1 contains a copy of the A9 and thus contains a nested sequence A3 ≤ A4 ≤ · · · ≤ A9. Shortly after Co1 was discovered, J. G. Thompson recognised that the normalizer of each of the groups in this sequence (apart from that of A8) is maximal in Co1 and the resulting collection of subgroups

· 3 Suz : 2, (A4 × G2(4)) : 2, (A5 × HJ) : 2, (A6 × U3(3)) : 2, (A7 × L2(7)) : 2,A8 × S4,A9 × S3 is now known as the Thompson chain, where Suz denotes the Suzuki simple group and HJ denotes the Hall-Janko group. Remarkably, we can start at the other end in the sense that if we consider U3(3) in a certain way we obtain . a construction which produces each of the groups U3(3) : 2, HJ : 2,G2(4) : 2, 3 Suz : 2, 2 × Co1 spontaneously. Indeed, a presentation containing a parameter n is given which, for n = 3, 4, 5, 6, 7 defines each of the above groups; n appears just twice in the presentation. Specifically, we associate with each directed edge ij of Kn (the complete graph on n vertices) an element −1 tij of order 7 in some group G, where tji = tij . We insist that G possesses automorphisms corresponding to the permuting the n vertices of our Kn, and in addition an automorphism which squares each of the tij. If we now factor by a relation which ensures that a triangle generates U3(3), then a K4 . generates HJ, a K5 generates G2(4), a K6 generates 3 Suz and a K7 generates Co1. What happens for n ≥ 8 will be explained fully in the talk. Thus this is not simply a sequence of nested subgroups in a larger group, but a finite family of closely-related perfect groups.

Meinolf Geck (Stuttgart)

Towards an atlas of generic character tables

We discuss the current state — including development, applications and open problems — of the CHEVIE project concerning generic character tables of finite groups of Lie type and related structures.

Sasha Ivanov (Imperial College)

The Majorana theory

The discoveries made within the theory in the title since its launch in April 2008 will be discussed together with suggestions for further directions of development.

Radha Kessar (City Univ. London)

On the block distribution of ordinary characters of finite quasi-simple groups of Lie type via Lusztig induction.

I will report on recent progress on the problem of describing the p-block distribution of the set of ordinary irreducible characters of a finite quasi-simple group (in characteristic different from p) in terms of the Lusztig labels of the irreducible characters. In particular, I will present joint work with G. Malle which gives a unified parametrisation of these p-blocks via Lusztig’s induction functor. This combines and builds upon earlier papers of Fong-Srinivasan, Schewe, Broue-Malle-Michel, Cabanes-Enguehard, Enguehard, Bonnafe- Rouquier and Kessar-Malle. Our results are specifically tailored for use in an inductive approach to the counting conjectures of modular representation theory.

Martin Liebeck (Imperial College)

Character ratios, growth and random walks on groups of Lie type

For a finite group G, a character ratio is a ratio of the form χ(g)/χ(1) where g is an element of G and χ is an irreducible character of G. These ratios occur natural in a variety of contexts, for example as the eigenvalues of various Markov processes associated with G. In the talk I will focus on a new result on character ratios, and describe some of its applications to the theory of growth and random walks on simple groups of Lie type, and also to representation varieties of various finitely presented groups.

Gunter Malle (TU Kaiserslautern)

Representation theory of finite groups of Lie type: Some open problems

We will discuss some basic open questions in the representation theory of finite nearly simple groups of Lie type and their relation to various conjectures in representation theory of arbitrary finite groups.

Gabriel Navarro (Valencia)

Character tables and Sylow subgroups

If G is a finite group and P is a Sylow p-subgroup of G, then there is information on the group P/P 0 in the character table of G.

Gabriele Nebe (RWTH Aachen)

Automorphisms of extremal codes

Extremal codes are self-dual binary codes with largest possible minimum distance. In 1973 Neil Sloane published a short note asking whether there is an extremal code of length 72. Since then many mathematicians search for such a code, developing new tools to narrow down the structure of its . We now know that, if such a code exists, then its automorphism group has order ≤ 5. The methods for studying this question involve explicit and constructive applications of well known classical theorems in algebra and group theory, for instance Conway’s and Pless’ application of Burnside’s orbit counting theorem and quadratic reciprocity dating back to the 1980’s. More recent and partly computational methods are based on representation theory of finite groups. The talk will survey some aspects of this ongoing search.

Simon Norton (Cambridge)

The Monster and the projective plane of order 3

At the time of publication of the ATLAS, work was well under way on understanding the description of the Monster simple group (or, to be precise, the Bimonster M o 2) in terms of a set of involutory generators corresponding to points and lines of the projective plane of order 3. This talk will describe what might have appeared in the ATLAS had this work been further advanced.

Eamonn O’Brien (Auckland)

Algorithms for linear groups: successes, failures, challenges

We review progress over the past 30 years on developing effective algorithms for linear groups. We identify some successes and failures, and various challenge problems.

Cheryl Praeger (Perth)

Conway groupoids and other structures Recently some colleagues (Nick Gill, Neil Gillespie and Jason Semeraro) and I have had some fun exploring some structures which are similar in spirit to the wonderful six-transitive subset M13 introduced by John Conway in 1997. Along the way we encountered regular two-graphs, three-transposition groups, and various kinds of designs. We thank John for the inspiration. I’ll attempt to explain what we found, and mention some of our unanswered questions.

Raphael Rouquier (UCLA)

What to represent finite groups on?

It would be desirable, for many problems in representation theory of finite groups, to use more rigid objects than vector spaces. I will explain the relevance of, and the limitation of, equivariant homotopy theory to local representation theory. Deligne-Lusztig theory provides a geometric approach to representation theory for finite groups of Lie type. I will discuss some of its applications to modular representation theory and to rationality properties via group actions on motives.

Alex Ryba (Queens College, CUNY)

Relative eigenvectors

A vector whose images under a pair of matrices are proportional is called a relative eigenvector. The problem of finding relative eigenvectors has a number of (computational) group theoretic applications. I will describe these applications, give an algorithm for finding relative eigenvectors and discuss the computational complexity of the problem and its extension to the case of many matrices.

Aner Shalev (Jerusalem)

Rapid expansion in finite simple groups

In recent years there has been intense interest in the expansion of powers of subsets of (nonabelian) finite simple groups, and remarkable results were proved. I will provide background and then focus on some new results – joint with Martin Liebeck and Gili Schul – establishing very fast 2-step expansion of normal subsets A of finite simple groups G. For example, we prove that for every  > 0 there exists δ > 0 such that if |A| ≤ |G|δ then |A2| ≥ |A|2−. Consequences and extensions will also be discussed, as well as a version for algebraic groups.

Ron Solomon (Ohio State)

What are the sporadic groups?

This talk will explore the possibility of an intrinsic (local) definition of a and a characterization theorem for a set of groups similar to the set tabulated in the Atlas. This includes work of Gorenstein, Lyons, Capdeboscq, Franchi, Mainardis and Lucido.

John Thompson (Cambridge)

(TBA)