  In an effort to better understand the formation of galaxy groups, we examine
the kinematics of a large sample of spectroscopically confirmed X-ray galaxy
groups in the Cosmic Evolution Survey (COSMOS) with a high sampling of galaxy
group members up to $z=1$. We compare our results with predictions from the
cosmological hydrodynamical simulation of {\sc Horizon-AGN}. Using a
phase-space analysis of dynamics of groups with halo masses of
$M_{\mathrm{200c}}\sim 10^{12.6}-10^{14.50}M_\odot$, we show that the brightest
group galaxies (BGG) in low mass galaxy groups ($M_{\mathrm{200c}}<2 \times
10^{13} M_\odot$) have larger proper motions relative to the group velocity
dispersion than high mass groups. The dispersion in the ratio of the BGG proper
velocity to the velocity dispersion of the group,
$\sigma_{\mathrm{BGG}}/\sigma_{group}$, is on average $1.48 \pm 0.13$ for low
mass groups and $1.01 \pm 0.09$ for high mass groups. A comparative analysis of
the {\sc Horizon-AGN} simulation reveals a similar increase in the spread of
peculiar velocities of BGGs with decreasing group mass, though consistency in
the amplitude, shape, and mode of the BGG peculiar velocity distribution is
only achieved for high mass groups. The groups hosting a BGG with a large
peculiar velocity are more likely to be offset from the $L_x-\sigma_{v}$
relation; this is probably because the peculiar motion of the BGG is influenced
by the accretion of new members.
