The mass function and distribution of velocity dispersions for UZC groups of galaxies

We measure the distribution of velocity dispersions of groups of galaxies identified in the UZC catalog; we use the distribution to derive the group mass function. We introduce a new method which makes efficient use of the entire magnitude limited catalog. Our determination of \nst includes a significant contribution from low luminosity systems that would be missing in a volume limited sample. We start from a model for the probability density function of the total number of group members and reproduce the observed distribution. We take several effects like the local fluctuations in volume density, limited sampling and group selection into account. We estimate the relation between total number of members, total luminosity and true velocity dispersion. We can then reproduce not only the observed distribution of \sv but also the distributions of the number of group members, the total groups luminosity, and virial mass. The best fit to the data in the true velocity dispersion range 100 km/s < \sigma < 750 km/s is a power law model with a slope -3.4 and a normalization (1.27 +/- 0.21) x 10^{-5} h^3 Mpc^{-3}. The predictions of CDM models are consistent with the mass function we derive from this distribution of velocity dispersions.