Unique factorization in invariant power series rings

Let G be a finite group, k a perfect field, and V a finite-dimensional kG-module. We let G act on the power series k V by linear substitutions and address the question of when the invariant power series k V G form a unique factorization domain. We prove that for a permutation module for a p-group in characteristic p, the answer is always positive. On the other hand, if G is a cyclic group of order p, k has characteristic p, and V is an indecomposable kG-module of dimension r with 1 r p, we show that the invariant power series form a unique factorization domain if and only if r is equal to 1, 2, p−1 or p. This contradicts a conjecture of Peskin.