  [Abridged] The relatively recent insight that energy input from supermassive
black holes (BHs) can have a substantial effect on the star formation rates
(SFRs) of galaxies motivates us to examine its effects on the scale of galaxy
groups. At present, groups contain most of the galaxies and a significant
fraction of the overall baryon content of the universe. To explore the effects
of BH feedback on groups, we analyse two high resolution cosmological hydro
simulations from the OverWhelmingly Large Simulations project. While both
include galactic winds driven by supernovae, only one includes feedback from
BHs. We compare the properties of the simulated groups to a wide range of
observational data, including hot gas radial profiles and gas mass fractions
(fgas), luminosity-mass-temperature (L-M-T) scaling relations, K-band
luminosity of the group and its central brightest galaxy (CBG), SFRs and ages
of the CBG, and gas/stellar metallicities. Both runs yield entropy profiles
similar to the data, while the run without AGN feedback yields highly peaked
temperature profiles, in discord with the observations. Energy input from BHs
significantly reduces fgas for groups with masses less than ~10^14 Msun,
yielding fgas-T and L-T relations that are in agreement with the data. The run
without AGN feedback suffers from the well known overcooling problem; the
resulting K-band luminosities are much larger than observed. By contrast, the
run that includes BH feedback yields K-band luminosities and CBG SFRs and ages
in agreement with current estimates. Both runs yield very similar gas-phase
metallicities that match X-ray data, but they predict very different stellar
metallicities. Based on the above, galaxy groups provide a compelling case that
BH feedback is a crucial ingredient in the formation of massive galaxies.
