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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-rayobservations from the XMM-Newton and Chandra observatories. We reach an X-rayflux limit of $3\times10^{-16}\;ergs\;cm^{-2}\;s^{-1}$ in 0.5--2 keV range, andidentify 247 X-ray groups with$M_{200c}=8\times10^{12}-3\times10^{14}\;M_{\odot}$ at a redshift range of$0.08\leq z<1.53$, using the multiband photometric redshift and the masterspectroscopic redshift catalogues of the COSMOS. The X-ray centres of groupsare determined using high-resolution Chandra imaging. We investigate therelations between the offset of the brightest group galaxies (BGGs) from haloX-ray centre and group properties and compare with predictions fromsemi-analytic models and hydrodynamical simulations. We find that BGG offsetdecreases with both increasing halo mass and decreasing redshift with no strongdependence on the X-ray flux and SNR. We show that the BGG offset decreases asa function of increasing magnitude gap with no considerable redshift dependenttrend. The stellar mass of BGGs in observations extends over a wider dynamicrange compared to model predictions. At $z<0.5$, the central dominant BGGsbecome more massive than those with large offsets by up to 0.3dex, in agreementwith model prediction. The observed and predicted lognormal scatter in thestellar mass of both low- and large-offset BGGs at fixed halo mass is$\sim0.3$dex.

Chandra centres for COSMOS X-ray galaxy groups: Differences in stellar properties between central dominant and offset brightest group galaxies