Mathematical Proceedings of the Cambridge Philosophical Society



Subgroups of simple algebraic groups containing elements of fundamental subgroups


MARTIN W. LIEBECK a1 and GARY M. SEITZ a2
a1 Imperial College, London SW7 2BZ, United Kingdom
a2 University of Oregon, Eugene, Oregon 97403, USA

Abstract

Let G be a simple algebraic group over an algebraically closed field K of characteristic p. If Σ is the root system of G and Uα is the root subgroup of G corresponding to a long root α[set membership]Σ, then [left angle bracket]Uα, U−α[right angle bracket] is an image of SL2 and any G-conjugate of this subgroup is called a fundamental subgroup of G. In [LS1], the closed connected semisimple subgroups of G generated by long root elements were determined. Of course, long root elements are unipotent elements of fundamental subgroups. In this paper we consider subgroups of G which are generated by semisimple elements lying in fundamental subgroups.

Our first three results (Theorems 1–3) concern semisimple connected subgroups of G which contain a maximal torus of a fundamental subgroup; we call such a torus a fundamental torus. Notice that for classical groups, the elements in fundamental tori have fixed spaces of small codimension in the natural module; indeed, for the groups SL(V) and Sp(V), the elements of fundamental tori are precisely those semisimple elements with fixed space of codimension 2, while for SO(V), the codimension is 4.

As consequences of these results, we obtain information on subgroups (finite or infinite) of classical groups which are generated by conjugates of a single element of a fundamental torus of order at least 5 (see Theorems 4, 5 and Corollary 6); for example, Theorem 5 determines those finite irreducible subgroups containing such an element which are quasisimple and of Lie type in characteristic p.

We now state our results in detail. Recall from [LS1] that a subsystem subgroup of G is a connected semisimple subgroup which is invariant under a maximal torus of G.

(Received October 29 1997)
(Revised April 2 1998)