  We analyze hierarchical structure formation based on scale-free initial
conditions in an Einstein-de Sitter universe, including a baryonic component.
We present three independent, smoothed particle hydrodynamics (SPH)
simulations, performed with two different SPH codes (TreeSPH and P3MSPH) at two
resolutions. Each simulation is based upon identical initial conditions, which
consist of Gaussian distributed initial density fluctuations that have an n=-1
power spectrum. The baryonic material is modeled as an ideal gas subject only
to shock heating and adiabatic heating and cooling. The evolution is expected
to be self-similar in time, and under certain restrictions we identify the
expected scalings for many properties of the distribution of collapsed objects
in all three realizations. The distributions of dark matter masses, baryon
masses, and mass and emission weighted temperatures scale quite reliably.
However, the density estimates in the central regions of these structures are
determined by the degree of numerical resolution. As a result, mean gas
densities and luminosities obey the expected scalings only when calculated
within a limited dynamic range in density contrast. The temperatures and
luminosities of the groups show tight correlations with the baryon masses,
which can be well-represented by power-laws. The Press-Schechter (PS)
approximation predicts the distribution of group dark matter and baryon masses
fairly well, though it tends to overestimate the baryon masses. Combining the
PS mass distribution with the measured relations for T(M) and L(M) predicts the
temperature and luminosity distributions reasonably, though there are some
discrepancies at high temperatures/luminosities. The three simulations agree
well for the properties of groups that are resolved by 32 or more particles.
