We study the role of environment in the evolution of central and satellite galaxies with the Sloan Digital Sky Survey. We begin by studying the size-mass relation, replicating previous studies, which showed no difference between the sizes of centrals and satellites at fixed stellar mass, before turning our attention to the size-core velocity dispersion ($\sigma_0$) and mass-$\sigma_0$ relations. By comparing the median size and mass of the galaxies at fixed velocity dispersion we find that the central galaxies are consistently larger and more massive than their satellite counterparts in the quiescent population. In the star forming population we find there is no difference in size and only a small difference in mass. To analyse why these difference may be present we investigate the radial mass profiles and stellar metallicity of the galaxies. We find that in the cores of the galaxies there is no difference in mass surface density between centrals and satellites, but there is a large difference at larger radii. We also find almost no difference between the stellar metallicity of centrals and satellites when they are separated into star forming and quiescent groups. Under the assumption that $\sigma_0$ is invariant to environmental processes, our results imply that central galaxies are likely being increased in mass and size by processes such as minor mergers, particularly at high $\sigma_0$, while satellites are being slightly reduced in mass and size by tidal stripping and harassment, particularly at low $\sigma_0$, all of which predominantly affect the outer regions of the galaxies.