We present relations between X-ray luminosity and velocity dispersion (L - σ), X-ray luminosity and gas mass (L - Mgas), and cluster radius and velocity dispersion (r500 - σ) for 62 galaxy clusters in the HIFLUGCS, an X-ray flux-limited sample minimizing bias toward any cluster morphology. Our analysis in total is based on ~ 1.3 Ms of clean X-ray XMM-Newton data and 13439 cluster member galaxies with redshifts. Cool cores are among the major contributors to the scatter in the L - σ relation. When the cool-core-corrected X-ray luminosity is used the intrinsic scatter decreases to 0.27 dex. Even after the X-ray luminosity is corrected for the cool core, the scatter caused by the presence of cool cores dominates for the low-mass systems. The scatter caused by the non-cool-core clusters does not strongly depend on the mass range, and becomes dominant in the high-mass regime. The observed L - σ relation agrees with the self-similar prediction, matches that of a simulated sample with AGN feedback disregarding six clusters with <45 cluster members with spectroscopic redshifts, and shows a common trend of increasing scatter toward the low-mass end, i.e., systems with σ ≤ 500 km s-1. A comparison of observations with simulations indicates an AGN-feedback-driven impact in the low-mass regime. The best fits to the L - Mgas relations for the disturbed clusters and undisturbed clusters in the observational sample closely match those of the simulated samples with and without AGN feedback, respectively. This suggests that one main cause of the scatter is AGN activity providing feedback in different phases, e.g. during a feedback cycle. The slope and scatter in the observed r500 - σ relation is similar to that of the simulated sample with AGN feedback except for a small offset but still within the scatter.
