Introduction to the subgroup structure of algebraic groups  pp. 129-150

Introduction to the subgroup structure of algebraic groups

By M. W. Liebeck

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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.