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 Horizon-AGN . Using a phase-space analysis of dynamics of groups with halo masses of M _ { \mathrm { 200 c } } \sim 10 ^ { 12.6 } -10 ^ { 14.50 } M _ { \odot } , we show that the brightest group galaxies ( BGG ) in low mass galaxy groups ( M _ { \mathrm { 200 c } } < 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 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 .