Edited by Roger W. Carter
Edited by Meinolf Geck
Publications of the Newton Institute (No. 16)
Publisher: Cambridge University Press
Print Publication Year: 1998
Online Publication Date:January 2010
Online ISBN:9780511600623
Hardback ISBN:9780521643252
Chapter DOI: http://dx.doi.org/10.1017/CBO9780511600623.008
Subjects: Algebra
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Introduction
The purpose of this article is to discuss various results concerning the subgroups of simple algebraic groups G and of the corresponding finite groups of Lie type GF (where F is a Frobenius morphism). There are five sections. The first contains some background on simple groups, automorphisms and reductive subgroups. In the second section we present material on two important classes of subgroups which contain a maximal torus of G: the parabolic subgroups, and the reductive “subsystem” subgroups. Section 3 contains a discussion of unipotent classes, and of subgroups of G containing various particular types of such elements. In section 4 we concentrate on closed subgroups of classical groups G. We present a recent reduction theorem which shows that any such subgroup either lies in a member of a class of naturally defined “geometric” subgroups of G, or is essentially a quasisimple group acting irreducibly on the natural module for G. Use of this result, together with a standard process involving Lang's theorem for linking finite and algebraic groups, yields a new proof of a well known reduction theorem of Aschbacher for finite classical groups, which we discuss. In the final section 5, we describe the picture for exceptional groups G. Again, there is a reduction theorem, reducing the study of subgroups H to the case where H is almost simple, and we sketch also the substantial body of recent results concerning the latter case.
pp. i-iv
pp. v-vi
pp. vii-viii
Introduction to algebraic groups and Lie algebras: Read PDF
pp. 1-20
Weyl groups, affine Weyl groups and reflection groups: Read PDF
pp. 21-40
Introduction to abelian and derived categories: Read PDF
pp. 41-62
Finite groups of Lie type: Read PDF
pp. 63-84
Generalized Harish-Chandra theory: Read PDF
pp. 85-104
Introduction to quantum groups: Read PDF
pp. 105-128
Introduction to the subgroup structure of algebraic groups: Read PDF
pp. 129-150
Introduction to intersection cohomology: Read PDF
pp. 151-172
An introduction to the Lusztig Conjecture: Read PDF
pp. 173-188
pp. 189-191