The Module Structure of a Group Action on a Polynomial Ring

For any representation of a p-group G on a vector space of dimension 3 over a finite field k of characteristic p, we show how the symmetric algebra, regarded as a kG-module, can be expressed as a direct sum of kG-modules, each one of which is isomorphic to a summand in low degree. It follows that, for any group G, only a finite number of isomorphism classes of summands can occur.