Kovacs, LStohr, Ralph2015-12-080021-8693http://hdl.handle.net/1885/30920Motivated 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.Keywords: Free groups; Free Lie algebras; Free metabelian Lie algebras; Relation modules201010.1016/j.jalgebra.2009.10.0072016-02-24