Asymptotic behaviour of Lie powers and Lie modules

Let $V$ be a finite-dimensional $FG$-module, where $F$ is a field of prime characteristic $p$ and $G$ is a group. We show that, when $r$ is not a power of $p$, the Lie power $L^r(V)$ has a direct summand $B^r(V)$ which is a direct summand of the tensor power $V^{\otimes r}$ and which satisfies $\dim B^r(V)/\dim L^r(V) \to 1$as$r \to \infty$\mathrm{.}Similarly,forthesamevaluesof$r$,weobtainaprojectivesubmodule$C®$oftheLiemodule$\Lie®$over$F$suchthat$\dim C®/\dim \Lie® \to 1$as$r \to \infty$.