Generation of Linear Groups

Let G be a finite, primitive subgroup of GL(V)= GL(n, D), where V is an n-dimensional vector space over the division ring D. Assume that G is generated by “nice” transformations. The problem is then to try to determine (up to GL(V)-conjugacy) all possibilities for G. Of course, this problem is very vague. But it is a classical one, going back 150 years, and yet very much alive today. The purpose of this paper is to discuss both old and new results in this area, and in particular to indicate some of its history. Our emphasis will be on especially geometric situations, rather than on representation-theoretic ones.