We show that the ratio between the stellar mass of central galaxy and the mass of its host halo, $f_c equiv M_{*,c}/M_{rm h}$, can be used as an observable proxy of halo assembly time, in that galaxy groups with higher $f_c$ assembled their masses ea
rlier. Using SDSS groups of Yang et al., we study how $f_c$ correlates with galaxy properties such as color, star formation rate, metallicity, bulge to disk ratio, and size. Central galaxies of a given stellar mass in groups with $f_c>0.02$ tend to be redder in color, more quenched in star formation, smaller in size, and more bulge dominated, as $f_c$ increases. The trends in color and star formation appear to reverse at $f_c<0.02$, reflecting a down-sizing effect that galaxies in massive halos formed their stars earlier although the host halos themselves assembled later (lower $f_c$). No such reversal is seen in the size of elliptical galaxies, suggesting that their assembly follows halo growth more closely than their star formation. Satellite galaxies of a given stellar mass in groups of a given halo mass tend to be redder in color, more quenched in star formation and smaller in size as $f_c$ increases. For a given stellar mass, satellites also tend to be smaller than centrals. The trends are stronger for lower mass groups. For groups more massive than $sim 10^{13}{rm M}_odot$, a weak reversed trend is seen in color and star formation. The observed trends in star formation are qualitatively reproduced by an empirical model based on halo age abundance matching, but not by a semi-analytical model tested here.
