We compare predictions of a number of empirical models and numerical simulations of galaxy formation to the conditional stellar mass functions (CSMF)of galaxies in groups of different masses obtained recently by Lan et al. to test how well different models accommodate the data. The observational data clearly prefer a model in which star formation in low-mass halos changes behavior at a characteristic redshift $z_csim 2$. There is also tentative evidence that this characteristic redshift depends on environment, becoming $z_csim 4$ in regions that eventually evolve into rich clusters of galaxies. The constrained model is used to understand how galaxies form and evolve in dark matter halos, and to make predictions for other statistical properties of the galaxy population, such as the stellar mass functions of galaxies at high $z$, the star formation and stellar mass assembly histories in dark matter halos. A comparison of our model predictions with those of other empirical models shows that different models can make vastly different predictions, even though all of them are tuned to match the observed stellar mass functions of galaxies.
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