Modular representations and almost free actions

Let k be a field of characteristic p > 0, and let G be a finite group whose order is divisible by p. We examine kG-modules V of large dimension with the property that the spaces of fixed points of non-trivial elements of G all have small dimension. It is shown that the existence of such a module imposes severe restrictions on the structure of G. 1991 Mathematics Subject Classification: 20D99, 20C20.