What aspects of galaxy environment matter?

We determine what aspects of the density field surrounding galaxies most affect their properties. For Sloan Digital Sky Survey galaxies, we measure the group environment, meaning the host group luminosity and the distance from the group center (hereafter, ``groupocentric distance''). For comparison, we measure the surrounding density field on scales ranging from 100 kpc/h to 10 Mpc/h. We use the relationship between color and group environment to test the null hypothesis that only the group environment matters, searching for a residual dependence of properties on the surrounding density. Generally, red galaxies are slightly more clustered on small scales (about 100--300 kpc/h) than the null hypothesis predicts, possibly indicating that substructure within groups has some importance. At large scales (> 1 Mpc/h), the actual projected correlation functions of galaxies are biased at less than the 5% level with respect to the null hypothesis predictions. We exclude strongly the converse null hypothesis, that only the surrounding density (on any scale) matters. These results generally encourage the use of the halo model description of galaxy bias, which models the galaxy distribution as a function of host halo mass alone. We compare these results to proposed galaxy formation scenarios within the Cold Dark Matter cosmological model