Are All Groups Finite?

Abstract

This paper is dedicated to Walter Feit on the occasion of his 65th birthday. Its contents were presented in part at the 1995 Ohio State finite group representation conference organized in celebration of that birthday. Primarily, the paper is a discussion of some classical and recent developments in the modular representation theory of finite groups of Lie type, and the problems which drive that theory. But there is also a philosophical thread . . . An old question which arose again at the conference is the following: Are all groups finite? That is, applications and broader issues aside, if we think only of our interest in finite group theory itself, is it possible to safely ignore other groups? My viewpoint is that the answer to this question has two parts: First, in representation theory, at least, we cannot ignore the infinite complex Lie groups and their characteristic p analogs, the algebraic groups over @,. The second part of my answer is that we can, nevertheless, hope to find understandings within finite group theory and finite dimensional algebra of ideas naturally suggested by these continuous contexts, and take them further. Let me begin by convincing you of the first part of my answer: Suppose one is considering a finite group G(F,) of Lie type, such as the special linear group SL(n, 4) of degree n with coefficients in the field F, of 4 elements, q a power of a prime p. The Classification of finite simple groups asserts that almost all of the latter are variations on the finite groups of Lie type together with the alternating groups. Much earlier (1963), Steinberg [341,[351 proved all irreducible representations of G(lF,) with coefficients in a finite field of characteristic p, and, thus, in the algebraic closure F,, come by restriction from the irreducible representations over @, of the algebraic group G(@,). The latter group is, of course, quite infinite. It is the analog via the Zariski topology, of the complex analytic Lie group G(C). Moreover, the representations we need are continuous, and even “analytic”, in the sense that they are locally defined by polynomial functions. Now, the theory of finite-dimensional