Recall that a prehomogeneous vector space, is a representation $V$ of a linear algebraic group $G$ having an open $G$-orbit. Let $Z$ be the neutral connected component of the stabilizer of a point of the open orbit. Assume that $G$ and $Z$ are both reductive. Is it always true that the canonical homomorphism $Z/[Z,Z]\to G/[G,G]$ has finite kernel? (This homomorphism is interesting because its cokernel is responsible for irreducible components of the complement of the open orbit)