Does the Complex Group Index Satisfy the Kramers-Kronig Relations in Linear Optics?

By combining the formula for the group complex refractive index with the Kramers-Kronig relations connecting the real and imaginary parts of the complex susceptibility in the context of linear optics, we rigorously prove the existence of the Kramers-Kronig relation for the imaginary part of the group refractive index, and non-existence of the relation for the real part. We briefly discuss the various theoretical implications this result.