  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.
