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.
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