Penrose limits of homogeneous spaces

We prove that the Penrose limit of a spacetime along a homogeneous geodesic is a homogeneous plane wave spacetime and that the Penrose limit of a reductive homogeneous spacetime along a homogeneous geodesic is a Cahen–Wallach space. We then consider several homogenous examples to show that these results are indeed sharp and conclude with a remark about the existence of null homogeneous geodesics.