Maschke's theorem says that every finite-dimensional representation of a finite group is completely reducible. Is there a simple example of an infinite-dimensional representation of a finite group which is not completely reducible?
EDIT: As mentioned in the answers, there is actually no finite-dimensional caveat in Maschke's theorem. It seems that I just got unlucky, in the that the first couple of references I found included a finite-dimensional assumption.
Best Answer
I was not able to find any reference for Maschke's theorem talking only about the finite dimensional representations.
The "modern" statement of Maschke's theorem (or at lest, the one ring theorists like) is this:
Semisimplicity, linked above, is the condition where all right $R$ modules split into sums of simple modules. This is the corresponding fact to the result in representation theory about all representations being completely reducible.
Now if $R$ is a field of characteristic zero, the order of $G$ is going to be a unit no matter what order it has, and furthermore, fields are semisimple. So in particular, you have a corollary:
When the order of $G$ divides the characteristic of a field $F$, $F[G]$ does have representations that are not completely reducible. The easiest example in that case would have to be $F[G]$ itself, which necessarily has a nonzero Jacobson radical.
As a toy example, you could take the cyclic group of order two $C_2=\{1,c\}$ and the field $F_2$ of order two, and consider its group algebra $F_2(C_2)$. You get a ring of four elements $\{0,1,c,1+c\}$. You can see that $(c+1)^2=0$, so that $\{0,c+1\}$ is the nilradical (which is the Jacobson radical in this case).