While in the hierarchical model of structure formation, groups of galaxies are believed to be a scaled-down version of clusters of galaxies, a similarity break 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 (L_X), the temperature (T) of hot diffuse gas, and the velocity dispersion (σ) of galaxies for groups and clusters of galaxies, based on the largest sample of 66 groups and 274 clusters drawn from the literature. Our best-fit L_X-T and L_X-σ relations for groups are given by L_X ∝ T~(5.57±1.79)∝ σ~(2.35±0.21), which deviates remarkably from those for clusters: L_X ∝ T~(2.79±0.08) ∝ σ~(5.30±0.21). The significance of these correlations has been justified by both the co-consistency test and the Kendall τ statistics. We have thus confirmed the existence of a similarity break in the L_X-T and L_(X-σ) relations between groups and clusters as claimed in previous work, although the best-fit σ-T relations remain roughly the same in the two systems: σ ∝ T~(0.64). Alternatively, the significant disagreement between the observationally fitted L_X-T and L_(X-σ) relations for groups and those expected from a perfect hydrostatic equilibrium hypothesis indicates that the X-ray emission of individual galaxies and the nongravitational 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.
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