Symmetric powers of modular representations for groups with a sylow subgroup of prime order

Let V he a representation of a finite group G over a field of characteristi c p. If p does not divide the group order, then Molien's formula gives the Hilbert series of the invariant ring. In this paper we find a replacement o f Molien's formula which works in the case that /G/ is divisible by p but n ot by p(2). We also obtain formulas which give generating functions encodin g the decompositions of all symmetric powers of V into indecomposables. Our methods can be applied to determine the depth of the invariant ring withou t computing any invariants. This leads to a proof of a conjecture of the se cond author on certain invariants of GL(2)(p), (C) 2001 Academic Press.