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∈Σ, then
〈Uα, U−α〉
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.