Factorisation of Lie resolvents

Let GG be a group, FF a field of prime characteristic pp, and VV a finite-dimensional FGFG-module. For each positive integer rr, the rrth homogeneous component of the free Lie algebra on VV is an FGFG-module called the rrth Lie power of VV. Lie powers are determined, up to isomorphism, by certain functions ΦrΦr on the Green ring of FGFG, called ‘Lie resolvents’. Our main result is the factorisation Φpmk=Φpm∘ΦkΦpmk=Φpm∘Φk whenever kk is not divisible by pp. This may be interpreted as a reduction to the key case of pp-power degree.