Upper panel: magnitude gap distribution of all groups (filled gray histogram) and groups selected according to the halo mass and BGG luminosity ranges of SI (solid red histogram), SII (dashed green histogram), and SIII (long-dashed blue histogram) at z = 0 as described in Sect. 2.2. Lower panel: BGG absolute r-band magnitude versus M₂₀₀ for all groups (gray points) at z = 0. We highlight the sub-sample of SI, SII, and SIII with red, green, and blue points, respectively. The fossil, control, and random groups are selected from these highlighted three sub-samples.