Factorisation of Lie resolvents

Let G be a group, F a field of prime characteristic p, and V a finite-dimensional FG-module. For each positive integer r, the rth homogeneous component of the free Lie algebra on V is an FG-module called the rth Lie power of V. Lie powers are determined, up to isomorphism, by certain functions Φr on the Green ring of FG, called ‘Lie resolvents’. Our main result is the factorisation Φpmk=Φpmring operatorΦk whenever k is not divisible by p. This may be interpreted as a reduction to the key case of p-power degree.