Skip to main content
edited tags
Link
Jim Humphreys
  • 52.9k
  • 4
  • 120
  • 240
Source Link
q.g.
  • 93
  • 4

A generalisation of the theorem of Maschke

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 $V=W\oplus W'$.