  Galaxy groups differ from clusters primarily by way of their lower masses,
M~10^14 M_sun vs. M~10^15 M_sun. We discuss how mass affects the thermal state
of the intracluster or the intragroup medium, specifically as to their entropy
levels and radial profiles. We show that entropy is produced in both cases by
the continuing inflow of intergalactic gas across the system boundary into the
gravitational potential well. The inflow is highly supersonic in clusters, but
weakly so in groups. The former condition implies strong accretion shocks with
substantial conversion of a large inflow kinetic into thermal energy, whereas
the latter condition implies less effective conversion of lower energies. These
features produce a conspicuous difference in entropy deposition at the current
boundary. Thereafter, adiabatic compression of the hot gas into the potential
well converts such time histories into radial profiles throughout a cluster or
a group. In addition, in both cases a location of the system at low z in the
accelerating universe or in a poor environment will starve out the inflow and
the entropy production, and produce flattening or even bending down of the
outer profile. We analyze in detail the sharp evidence provided by the two
groups ESO 3060170 and RXJ1159+5531 that have been recently observed in X rays
out to their virial radii, and find a close and detailed match with our
expectations.
