Is this kind of module possible? I have a homework in which i must give an example or prove its not. I can do neither.
We know the following about a reducible unfaithful module: $$ V = U_1 \oplus U_2 \oplus \dots \oplus U_n, $$ where $U_i$ is irreducible for some $v_i \in V$, and some $g, h \in G$, $g \cdot v_i = h \cdot v_i$. Any vector in $V$ can be written as a unique sum of vectors in $U_i$.
The uniqueness seems like it might give a contradiction, but I simply can't think of one, try as I might.
Any pointers would be greatly appreciated.