We use the galaxy stellar mass and halo merger tree information from the semi-analytic model galaxy catalogue of Font et al. (2009) to examine the accretion of galaxies into a large sample of groups and clusters, covering a wide range in halo mass (10E12.9 to 10E15.3 Msun/h), and selected from each of four redshift epochs (z=0, 0.5, 1.0 and 1.5). We find that clusters at all examined redshifts have accreted a significant fraction of their final galaxy populations through galaxy groups. A 10E14.5 Msun/h mass cluster at z=0 has, on average, accreted ~ 40% of its galaxies (Mstellar > 10E9 Msun/h) from halos with masses greater than 10E13 Msun/h. Further, the galaxies which are accreted through groups are more massive, on average, than galaxies accreted through smaller halos or from the field population. We find that at a given epoch, the fraction of galaxies accreted from isolated environments is independent of the final cluster or group mass. In contrast, we find that observing a cluster of the same halo mass at each redshift epoch implies different accretion rates of isolated galaxies, from 5-6 % per Gyr at z=0 to 15% per Gyr at z=1.5. We find that combining the existence of a Butcher Oemler effect at z=0.5 and the observations that galaxies within groups display significant environmental effects with galaxy accretion histories justifies striking conclusions. Namely, that the dominant environmental process must begin to occur in halos of 10E12 -- 10E13 Msun/h, and act over timescales of > 2 Gyrs. This argues in favor of a mechanism like strangulation, in which the hot halo of a galaxy is stripped upon infalling into a more massive halo . This simple model predicts that by z=1.5 galaxy groups and clusters will display little to no environmental effects.
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