Let G1 be a finite group of Lie
type defined over a field of characteristic p. The results of this paper represent an
attempt to achieve a better understanding of the subgroup structure of G1. It is
somewhat surprising how limited our knowledge is, in this regard. For example, while
centralizers of semisimple elements (i.e. p′-elements) of G1 have been studied in
detail and are fairly well understood, very little has been written about subgroups of
G1 generated by such centralizers. Even in explicit examples the analysis of such
subgroups can be very difficult, the difficulty stemming from an inability to
relate the generated group to the Lie structure of G1. To deal with these
situations and others we set up a framework that allows us to effectively study a
fairly large class of subgroups of G1 (those containing a maximal torus), by
studying subgroups of the corresponding algebraic group. Essential to the
development is a theory of root subgroups for arbitrary maximal tori of
G1.
