  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 $3\times10^{-16}\;ergs\;cm^{-2}\;s^{-1}$ in 0.5--2 keV range, and
identify 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 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.
