  Let G be a group, F a field of prime characteristic p and V a
finite-dimensional FG-module. Let L(V) denote the free Lie algebra on V
regarded as an FG-submodule of the free associative algebra (or tensor algebra)
T(V). For each positive integer r, let L^r(V) and T^r(V) be the rth homogeneous
components of L(V) and T(V), respectively. Here L^r(V) is called the rth Lie
power of V. Our main result is that there are submodules B_1, B_2, ... of L(V)
such that, for all r, B_r is a direct summand of T^r(V) and, whenever m \geq 0
and k is not divisible by p, $$ L^{p^mk}(V) = L^{p^m}(B_k) \oplus
L^{p^{m-1}}(B_{pk}) \oplus ... \oplus L^p(B_{p^{m-1}k}) \oplus L^1(B_{p^mk}).
$$ Thus every Lie power is a direct sum of Lie powers of p-power degree. The
approach builds on an analysis of T^r(V) as a bimodule for G and the Solomon
descent algebra.
