Actions of Linear Groups

Let G be a permutation group on a finite set X. A subset B of X is said to be a base for G if its pointwise stabilizer in G is just the identity. Bases have been studied since the early years of permutation group theory, particularly in connection with orders of primitive groups and, more recently, with computational group theory. The latter connection has arisen because a group element is completely determined by its effect on a base - so the existence of a small base allows one to store group elements efficiently.

There is a great deal of recent theory on bases of primitive permutation groups. Some of the focus has been on two major conjectures in the area: the Cameron-Kantor conjecture, which (roughly stated) says that a vast collection of types of primitive groups have bases of bounded size; and the Pyber conjecture, which relates the size of a minimal base of a primitive group to the order of the group in a remarkably close way. This proposal has more of the flavour of the former conjecture. Namely, for linear groups -- that is, subgroups of GL(n,q) for some dimension n and field GF(q), acting as a permutation group on the vectors of the underlying vector space V(n,q) -- we aim to determine the groups that have "small" bases; here, "small" initially will mean "size 2" (the smallest interesting size for a base), but will be extended to larger sizes as the project progresses. Clearly GL(n,q) itself does not have a small base unless n is small, but many of its subgroups do, and the aim is to classify these in as precise a way as possible. There is a substantial literature on this problem alone, but many new avenues to explore. An initial case will be to study the subgroups of GL(n,q) that are themselves simple groups of Lie type acting irreducibly on the underlying vector space V(n,q). Doing this will involve detailed use of the structure and representations of such simple groups, itself a very big subject.