Understanding the interaction between galaxies and their surroundings is central to building a coherent picture of galaxy evolution. Here we use Galaxy Evolution Explorer imaging of a statistically representative sample of 23 galaxy groups at z ≈ 0.06 to explore how local and global group environments affect the UV properties and dust-corrected star formation rates (SFRs) of their member galaxies. The data provide SFRs out to beyond 2R in all groups, down to a completeness limit and limiting galaxy stellar mass of 0.06 M⊙ yr and 1 × 10 M⊙, respectively. At fixed galaxy stellar mass, we find that the fraction of star-forming group members is suppressed relative to the field out to an average radius of R ≈ 1.5 Mpc ≈ 2R , mirroring results for massive clusters. For the first time, we also report a similar suppression of the specific SFR within such galaxies, on average by 40% relative to the field, thus directly revealing the impact of the group environment in quenching star formation within infalling galaxies. At fixed galaxy density and stellar mass, this suppression is stronger in more massive groups, implying that both local and global group environments play a role in quenching. The results favor an average quenching timescale of ≳ 2 Gyr and strongly suggest that a combination of tidal interactions and starvation is responsible. Despite their past and ongoing quenching, galaxy groups with more than four members still account for at least ∼25% of the total UV output in the nearby universe.