TITLE: A generalisation of the theorem of Maschke
QUESTION [9 upvotes]: 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 $V=W\oplus W'$.

REPLY [4 votes]: I. Schur explicitly proved that every periodic subgroup of ${\rm GL}(n,\mathbb{C})$ is completely reducible ( Corollary 36.3 in Curtis and Reiner's "Representation Theory of Finite Groups and Associative Algebras", Wiley and Sons, 1962), improving earlier results of Burnside,