tokens: may marginally agree with observations within errors. Figure 10. The SFR/ M s – z relation for S-I (left-hand panel), and for S-II to S-V (right-hand panel). The solid, dashed, and dotted lines show the median trends for BGGs in observations and in the SAMs of DLB07 and G11 , respectively. The grey point shows the BGG sSFR versus redshift in observations. In Fig. 11 , we show the median sSFR dependence of BGGs to the total mass of their host groups in observations (black points) and in the SAMs of G11 (red diamonds) and DLB07 (green squares), respectively. For S-I (left-hand panel), the median sSFR rapidly decreases as a function of increasing halo mass (the best-fitting linear relations corresponds to: SFR/ M s = (−4.4 ± 1.0) × M 200 + (3.8 ± 0.8). The observed trend is steeper than the model predictions. In agreement with the SAM predictions, we find no relationship between sSFR and M 200 for S-II to S-V over 0.1 z 1.3, as shown in the right-hand panel in Fig. 11 . These results suggest that, in the group scale haloes, sSFR of BGGs is a function of redshift with no halo mass dependence. Figure 11. The SFR/ M s – M 200 relation for S-I (left-hand panel), and for S-II to S-V (right-hand panel). The solid, dashed, and dotted lines show the median trends for BGGs in observations and in the SAM of DLB07 and G11 , respectively. The grey point shows the BGG sSFR versus redshift in observations. Only, for BGGs within S-I we find that the sSFR decreases as a function of halo mass. 5 SUMMARY AND CONCLUSIONS We select the BGGs in a well-defined X-ray-selected sample of galaxy groups from the COSMOS, XMM –LSS, and AEGIS fields. This sample covers a redshift range of 0.04 z 1.3 and contains about 407 galaxy groups with a mass ranging from 10 12.8 to 10 14 M ⊙ , the largest sample for such studies in the X-ray galaxy group regime. The sample allows us to study the distribution and evolution of the stellar mass and SFR of BGGs and the relation among the stellar mass, SFR, and halo mass. We compare our results with sample drawn from the Millennium simulations with three SAMs ( G11 , DLB07 , and B06 ). We summarize our results as follows. The shape of the stellar mass distribution of the BGGs evolves towards a normal distribution as the groups evolve. This evolution is quantified by the skewness and Kurtosis of the stellar mass distribution and their redshift dependence. At all redshifts, we find that the stellar mass distribution tends to skew towards low masses. The observed fraction of the BGGs shows a strong deviation from the Gaussian fit at ≈10 10.2 M ⊙ , which are generally SF/young population. In contrast, the SAM predictions show no redshift evolution of the stellar mass distribution. Within the probed SAMs, the shape of the stellar mass distribution predicted by B06 model is more consistent with the observations at z 0.5. We show that the average stellar mass of the BGGs evolves with redshift by a factor of ∼2 from z = 1.3 to the present day. At z
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