Based on a large sample of 254 220 galaxies in 81 089 groups, which are selected from the spectroscopic galaxy sample of the SDSS DR12, we investigate the radial distribution of incidences, morphologies, environmental densities, and star formation properties of the active galactic nucleus (AGN) host galaxies and star-forming galaxies (SFGs) in the groups at z<0.2, as well as their changes with group richness ($N_{rm rich}$). It is found that AGN fraction slightly declines with richness for the groups/clusters. The SFG fraction is on average about 2 times larger than the AGN fraction, with a significant declining trend with richness. The group AGNs are preferentially reside in spheroidal and bulge-dominated disc galaxies, whereas the majority of SFGs are late-type discs. Compared with the SFGs, the AGNs in poor groups ($5 leqslant N_{rm rich} leqslant 10$) are closer to group center. The AGN fraction does not change with the distance to the group center, whereas the SFG fraction tends to be higher in the outskirts. The AGNs in groups have a higher incidence than the SFGs for the massive ($log(M_*/M_{odot}) > 10.7$) galaxies, and the mean SFG fraction is about 6 times as that of AGNs in the late-type galaxies with lower masses at larger radius. The distribution of environmental luminosity densities shows that the AGNs are likely to be reside in a denser environment relative to the SFGs. Compared with the SFGs in groups, the group AGNs are found to have a higher mean stellar mass, a lower mean star formation rate, and an older mean stellar age.
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