1. In reference [27] below, it is shown that any embedding S(q) < G(K) of a finite group of Lie type S(q) in a simple algebraic group G(K) extends to an embedding S(K) < G(K) of algebraic groups, provided the field size q is reasonably large. This result has major consequences for subgroup structure, and it enables one to use material on algebraic subgroups to deduce results about finite subgroups. An important task is to reduce the bounds on the field size q in the above result.
  2. In the study of actions of algebraic groups on algebraic varieties, questions often arise concerning the fixed point spaces of elements or subgroups of the algebraic group in question. It should be possible to prove some general results in this area for actions of simple algebraic groups.
  3. All simple algebraic groups are generated by a collection of so-called fundamental subgroups, which are subgroups SL(2) defined in terms of the root system of the algebraic group in question. Many papers have been written on the structure of subgroups which contain certain elements of these fundamental subgroups; probably the most studied are the root elements, which are unipotent elements in such SL(2) subgroups. This theory should be extended to corresponding questions about subgroups containing elements of more general types of SL(2) subgroups; for instance those lying in a product of two commuting fundamental subgroups.