  We confirm that the standard assumption of isothermal, shock-heated gas in
cluster potentials is unable to reproduce the observed X-ray luminosity-
temperature relation of groups of galaxies. As an alternative, we construct a
physically motivated model for the adiabatic collapse of pre-heated gas into an
isothermal potential that improves upon the original work of Kaiser (1991). The
luminosity and temperature of the gas is calculated, assuming an appropriate
distribution of halo formation times and radiation due to both bremsstrahlung
and recombination processes. This model successfully reproduces the slope and
dispersion of the luminosity-temperature relation of galaxy groups. We also
present calculations of the temperature and luminosity functions for galaxy
groups under the prescription of this model. This model makes two strong
predictions for haloes with total masses M<10^13 M_sun, which are not yet
testable with current data: (1) the gas mass fraction will increase in direct
proportion to the halo mass; (2) the gas temperature will be larger than the
virial temperature of the mass. The second effect is strong enough that group
masses determined from gas temperatures will be overestimated by about an order
of magnitude if it is assumed that the gas temperature is the virial
temperature. The entropy required to match observations can be obtained by
heating the gas at the turnaround time, for example, to about 3 X 10^6 K at
z=1, which is too high to be generated by a normal rate of supernova
explosions. This model breaks down on the scale of low mass clusters, but this
is an acceptable limitation, as we expect accretion shocks to contribute
significantly to the entropy of the gas in such objects.
