First of all I am far from being an expert in representation theory, so it is possible (likely) that the following question is trivial (in fact a trivial reference question):
Let $\Gamma$ be a, let's say finite (abelian) group, and $R$ be an effective representation of $\Gamma$.
Is it true, that every irreducible $\Gamma$-representation is a subrepresentation of the $k$-th symmetric power representation $S^kR$, for some $k=0,1,\dots$ ?
(Note that it is enough to show that any regular representations of $\Gamma$ is a subrepresentation of $S^kR$.)
I strongly suspect that this is a well know fact in representation theory, but I couldn't find any reference so far and would therefore appreciate any kind of literature reference.