We present techniques for identifying and analyzing galaxy groups and apply them to large-scale particle-mesh (PM) N-body simulations of structure formation in three Ω0 = 1 cosmological models: cold plus hot dark matter (CHDM), with Ωcold = 0.6, Ων = 0.3, and Ωbaryon = 0.1 at bias b≡σ-18=1.5 and two cold dark matter (CDM) models, at bias b = 1.5 and b = 1.0. Groups are identified with the adaptive friends-of-friends algorithm of Nolthenius. Our most important conclusions follow. The standard group M/L method gives Ω0 ~= 0.08 for the CfA1 survey (for redshift link parameter V5 = 350), and, applied to our Ω0 = 1 simulations, it gives Ω0 ~= 0.12 for CHDM (V5 = 350) and Ω0 ~= 0.35 for CDM (V5 = 600). This Ω bias appears to be even stronger at higher resolution. We show quantitatively how three different effects conspire to produce this large discrepancy, and we conclude that low observed Ω values need not argue for a low-Ω universe. Our preferred statistics of groups show promise in becoming powerful discriminators between Gaussian cosmological models, whose Ων differ and are robust against several methods for assigning luminosity to dark matter halos, and for merging CfA1 data. However, our latest results at higher resolution show such strong sensitivity to how massive overmergers are broken up that more reliable ways of identifying luminous galaxies within large-scale simulations will be necessary before these statistics can provide reliable discrimination. When overmergers are broken up, the median virial-to-DM mass Mvir/MDM of three-dimensional-selected groups is ~1 for all simulations. Groups with MDM > 1014 M⊙ appear virialized in all simulations. We measure global (not pairwise) velocity biases bv, similar to previous studies. Within three-dimensional-selected groups, CHDM and CDM with b = 1.5 show a stronger bias of bv = 0.7-0.8, while CDM with b = 1.0 shows groups of bv ~= 1.