ABSTRACT Galaxy groups differ from clusters primarily by way of their lower masses, M ∼ 1014 M ⊙ versus M ∼ 1015 M ⊙. 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 bulk 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.