- There are many interesting problems in this area. Here is
a selection:
- 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.
- 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.
- 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.