First Author: Dave Benson email: /b\e/n\s/o\n/d\j (without the slashes) at math dot uga dot edu Address: Department of Mathematics, University of Georgia Second Author: Peter Webb email: webb@math.umn.edu Address: School of Mathematics, University of Minnesota, Minneapolis MN 55455 Title: Unique factorization in invariant power series rings Abstract: Let G be a finite group, k a perfect field of characteristic p, and Va 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, the answer is always positive. On the other hand, if G is a cyclic group of order p and V is an indecomposable kG-module of dimension r with 1 \le r \le 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. Preprint: April 2005. Updated: January 2006