It is well known that complete reducibility fails for finite groups if the characteristic divides the order of the group. We show that one recovers a version of Maschke's theorem for small enough representations in a very explicit way. This answers a conjecture of Serre on the vanishing of the first cohomology group. The methods depend upon knowing low dimensional representations. We will discuss methods for studying this and give another application to a conjecture of Larsen about decompositions of tensor products of representations (both a characteristic zero version and a modular version).