Factorisation of Lie resolvents

Let G be a group, F a field of prime characteristic p, and V a finite-dimensional F G-module. For each positive integer r, the rth homogeneous component of the free Lie algebra on V is an F G-module called the rth Lie power of V. Lie powers are determined, up to isomorphism, by certain functions Φr on the Green ring of F G, called 'Lie resolvents'. Our main result is the factorisation Φpm k = Φpm {ring operator} Φk whenever k is not divisible by p. This may be interpreted as a reduction to the key case of p-power degree. © 2006 Elsevier Ltd. All rights reserved.