  Given a faithful finite-dimensional representation $V$ of a finite group $G$
over any field $\mathbb{F}$, we show that any irreducible
${\mathbb{F}}G$-module $W$ appears, as a submodule or a quotient, in
$\mathrm{Sym}^m(V)$ for some integer $1 \leq m \leq |G|$ (that may depend on
$W$).
