Lie powers of relation modules for groups

Motivated by applications to abstract group theory, we study Lie powers of relation modules. The relation module associated to a free presentation G=F/N of a group G is the abelianization Nab=N/[N,N] of N, with G-action given by conjugation in F. The degree n Lie power is the homogeneous component of degree n in the free Lie ring on Nab (equivalently, it is the relevant quotient of the lower central series of N). We show that after reduction modulo a prime p this becomes a projective G-module, provided n>1 and n is not divisible by p.