Galaxy groups and clusters and their brightest galaxies within the cosmic web

Context. The evolution of galaxy groups and the brightest group galaxies (BGGs) is influenced by their location in the cosmic web.Aims. Our aim is to combine data on galaxy groups, their BGGs, and their location in the cosmic web, to determine classes of groups and clusters, and to obtain a better understanding of their properties and evolution.Methods. Data on groups and their BGGs are based on the Sloan Digital Sky Survey DR10 MAIN spectroscopic galaxy sample in the redshift range 0.009 <= z <= 0.200. We characterize the group environments by the luminosity-density field and their filament membership. We divide BGGs according to their star formation properties as quenched (Q), red star-forming galaxies (RSF), and blue star-forming galaxies (BSF). We apply multidimensional Gaussian mixture modelling to divide groups based on the properties of the groups, their BGGs, and their environments. We analyse the offset of BGGs with respect to the group centre, and the relation between the stellar velocity dispersion of BGGs sigma(star) and the group velocity dispersions sigma(v). For comparison we also analyse the properties of single galaxies of different star formation properties in various environments.Results. The galaxy groups in our sample can be divided into two main classes: high-luminosity rich groups and clusters, and low-luminosity poor groups with threshold luminosity L-gr(thr) = 15 x 10(10) h(-2) L-circle dot and total mass M-gr(thr) approximate to 23 x 10(12) h(-1) M-circle dot. The brightest galaxies in clusters and groups have different star formation properties. In rich groups and clusters approximate to 90% of the BGGs are red quenched galaxies, while in poor groups only approximate to 40 - 60% of BGGs are red and quenched, and the rest of the BGGs are star-forming, either blue (20 - 40% of BGGs) or red (similar to 17% of BCGs). Rich groups and clusters are located in global high-density regions (superclusters) in filaments or filament outskirts, while poor groups reside everywhere in the cosmic web regardless of the global density (superclusters or voids). Clusters with quenched BGGs have higher luminosities and their BGGs are closer to the cluster centre than in clusters with star-forming BGGs. Groups of the same richness with red (quenched and star-forming) BGGs are more luminous, and they lie in higher global density environment than groups with blue star-forming BGGs.Conclusions. Our results suggest that the evolution of groups and clusters and their BGGs is related to their location in the cosmic web. We emphasize the role of global high-density regions-superclusters as a special environment for group growth. The processes that shape the properties of groups and their BGG are different and/or have different timescales in groups and clusters.