The Lx-T, Lx-sigma and sigma-T relations for groups and clusters of   galaxies
  While in the hierarchical model of structure formation, groups of galaxies are believed to be the scaled-down version of clusters of galaxies, a similarity breaking in the fundamental laws may occur on the group scale, reflecting a transition between galaxy-dominated and intracluster medium dominated properties. In this paper, we present an extensive study of the relations between the X-ray luminosity (Lx), the temperature (T) of hot diffuse gas and the velocity dispersion (sigma) of galaxies for groups and clusters of galaxies, based on the largest sample of 66 groups and 274 clusters drawn from literature. Our best fit Lx-T and Lx-sigma relations for groups read Lx propto T^(5.57+-1.79) propto sigma^(2.35+-0.21), which deviates remarkably from those for clusters: Lx propto T^(2.79+-0.08) propto sigma^(5.30+-0.21). The significance of these correlations have been justified by both the co-consistency test and the Kendall's tau statistics. We have thus confirmed the existence of similarity breaking in the Lx-T and Lx-sigma relations between groups and clusters as claimed in previous work, although the best fit sigma-T relations remain roughly the same in both systems: sigma propto T^0.64. Alternatively, the significant disagreement between the observationally fitted Lx-T and Lx-sigma relations for groups and those expected from a perfect hydrostatic equilibrium hypothesis indicates that the X-ray emission of individual galaxies and the non-gravitational heating must play a potentially important role in the dynamical evolution of groups. Therefore, reasonable caution should be exercised in the cosmological applications of the dynamical properties of groups. 
Yan-Jie Xue and Xiang-Ping Wu