We present the results of a search for galaxy clusters and groups in the $sim2$ square degree of the COSMOS field using all available X-ray observations from the XMM-Newton and Chandra observatories. We reach an X-ray flux limit of $3times10^{-16};ergs;cm^{-2};s^{-1}$ in 0.5--2 keV range, and identify 247 X-ray groups with $M_{200c}=8times10^{12}-3times10^{14};M_{odot}$ at a redshift range of $0.08leq z<1.53$, using the multiband photometric redshift and the master spectroscopic redshift catalogues of the COSMOS. The X-ray centres of groups are determined using high-resolution Chandra imaging. We investigate the relations between the offset of the brightest group galaxies (BGGs) from halo X-ray centre and group properties and compare with predictions from semi-analytic models and hydrodynamical simulations. We find that BGG offset decreases with both increasing halo mass and decreasing redshift with no strong dependence on the X-ray flux and SNR. We show that the BGG offset decreases as a function of increasing magnitude gap with no considerable redshift dependent trend. The stellar mass of BGGs in observations extends over a wider dynamic range compared to model predictions. At $z<0.5$, the central dominant BGGs become more massive than those with large offsets by up to 0.3dex, in agreement with model prediction. The observed and predicted lognormal scatter in the stellar mass of both low- and large-offset BGGs at fixed halo mass is $sim0.3$dex.
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