Let G be a finite group acting on a finite dimensional vector space V. Let C be a nontrivial subspace of V. Let H be the subgroup of G that fixes C pointwise (the stabilizer of C). I'm fairly sure that H has to be abelian. The argument going along the lines that C contains simultaneous eigenvectors of all the elements of H so these must commute...

I have two questions :

(1)Is the above correct? Is there a reference with a derivation.

(2)Is there a more general statement or setting for this?

Thanks.