This should be a simple question, but i'm having a rather hard time though finding a explicit definition of a completely reducible group representation. Is it right to say that a presentation is completely reducible if the vector space on which the group $G$ is represented can be written as a direct sum of $G$-invariant subspaces?
Because if the answer is "yes", can't I say that every presentation is completely reducible, because it is the direct sum of the full vectorspace with the nullspace which are both $G$-invariant?
I wonder about this, because Mashke's theorem is formulated in my book as follows:
Every finite-dimensional representation of a finite group $G$ is completely reducible as the direct sum of irreducible representations.
But in the proof it's mentioned that $V$ itself can be irreducible (but they make no problem out of it), but by the theorem $V$ would then be irreducible and completely reducible at the same time. And if $V$ is completely irreducible, isn't any representation completely reducible then?