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 Φ^r on the Green ring of FG, called ‘Lie resolvents’. Our main result is the factorisation Φ^pmk=Φ^pm°Φ^k whenever k is not divisible by p. This may be interpreted as a reduction to the key case of p-power degree.