The theorem of Maschke tells us that every representation of a finite group is the direct sum of irreducible representation. More precisely:


Let $G$ be a finite group, $K$ a field whose characteristic does not divide the order of $G$, $V$ a $K$-vector space of finite dimension and $\rho:G\to {\rm GL}(V)$ a representation of $G$. If $W$ is a $G$-invariant vector space of $V$, then there exists a $G$-invariant complement $W'$ such that $V=W\oplus W'$. 


Now, I would like to prove or to find a counter-example to the following generalisation:


Let $G$ be a group **such that every element has finite order**, $K$ a field of characteristic $0$, $V$ a $K$-vector space of finite dimension and $\rho:G\to {\rm GL}(V)$ a representation of $G$. If $W$ is a $G$-invariant vector space of $V$, then there exists a $G$-invariant complement $W'$ such that $V=W\oplus W'$. 



A generalisation of the theorem of Maschke
Your expected generalization is true. Without loss of generality, you may assume that $G$ is a subgroup of $GL(V)$. A linear group whose elements have finite orders is locally finite, that's an old Theorem by Schur, and contained for instance in Wehrfritz's book *Infinite Linear Groups*. Now take a maximal linearly independent subset $S$ of $G$, so $\langle S\rangle$ and $G$ have the same invariant subspaces on $V$. By Schur, $\langle S\rangle$ is finite, so your generalization holds by Maschke.


