Let $G$ be a finite group, $p$ a prime number. We denote by $\mathbb{F}\_p$ the field of cardinality $p$. Let $V$ be an infinite dimensional representation of $G$ over $\mathbb{F}\_p$.


Must there be $G$-invariant, proper subspaces $U,W \leq V$ such that $U + W = V$?


I do not require the sum to be direct. The question should be equivalent to asking:


Must $V$ have a nontrivial decomposable image?


I am able to prove this if $p$ does not divide $|G|$ by applying Maschke's theorem to decompose $V$ into a direct sum of finite-dimensional irreducible subrepresentations. Even if in general the answer is negative, I would like to know about additional cases in which the conclusion holds, to say:


Under what conditions on $G$ can we find such subspaces?


Interesting cases can be abelian, solvable or any other "nice" classes of groups.



Decomposing representations of finite groups
I am not sure that I follow Rickard's argument, but here is a direct proof. Given an infinite dimensional module $V$ (over an arbitrary field.) for a finite group $G$, I want to argue first that there exists a proper $G$-submodule $U$ with finite codimension. Let $X < V$ be an arbitrary proper subspace with finite codimension. Then
$U = \bigcap\_{g \in G} X^g$ is $G$-invariant and is an intersection of finitely many proper subspaces with finite codimension, so is proper and has finite codimension. Next, let
$Y \subseteq V$ be a finite dimensional subspace such that $U + Y = V$. Let
$W = \sum\_{g \in G} Y^g$. Then $W$ is $G$-invariant and finite dimensional, so $W < V$. Also, $U + W = V$, as wanted.


