This is a crosspost from [math.SE][1].  Suppose $G$ and $H$ are discrete groups.  Is it always the case that any finite dimensional complex representation of $G\times H$ is of the form
$$
\bigoplus_i V_i \otimes W_i,
$$
where $V_i, W_i$ are reps of $G$ and $H$, respectively?

I know this is true when $G$ and $H$ are finite and when the representation of $G\times H$ is completely reducible, but is there a simple counterexample to the general case?  

I'm also curious if it is ``usually true," in some sense, that any rep of $G\times H$ has the above form.


  [1]: http://math.stackexchange.com/questions/413888/when-is-rg-times-h-rg-otimes-rh